mathlib documentation

data.set.basic

Basic properties of sets #

Sets in Lean are homogeneous; all their elements have the same type. Sets whose elements have type X are thus defined as set X := X → Prop. Note that this function need not be decidable. The definition is in the core library.

This file provides some basic definitions related to sets and functions not present in the core library, as well as extra lemmas for functions in the core library (empty set, univ, union, intersection, insert, singleton, set-theoretic difference, complement, and powerset).

Note that a set is a term, not a type. There is a coercion from set α to Type* sending s to the corresponding subtype ↥s.

See also the file set_theory/zfc.lean, which contains an encoding of ZFC set theory in Lean.

Main definitions #

Notation used here:

Definitions in the file:

Notation #

Implementation notes #

Tags #

set, sets, subset, subsets, image, preimage, pre-image, range, union, intersection, insert, singleton, complement, powerset

Set coercion to a type #

theorem has_subset.subset.trans {α : Type u_1} [has_subset α] [is_trans α has_subset.subset] {a b c : α} (h : a b) (h' : b c) :
a c
theorem has_subset.subset.antisymm {α : Type u_1} [has_subset α] [is_antisymm α has_subset.subset] {a b : α} (h : a b) (h' : b a) :
a = b
theorem has_ssubset.ssubset.trans {α : Type u_1} [has_ssubset α] [is_trans α has_ssubset.ssubset] {a b c : α} (h : a b) (h' : b c) :
a c
theorem has_ssubset.ssubset.asymm {α : Type u_1} [has_ssubset α] [is_asymm α has_ssubset.ssubset] {a b : α} (h : a b) :
¬b a
@[instance]
def set.has_le {α : Type u_1} :
has_le (set α)
Equations
@[instance]
def set.has_lt {α : Type u_1} :
has_lt (set α)
Equations
@[simp]
theorem set.top_eq_univ {α : Type u_1} :
@[simp]
theorem set.bot_eq_empty {α : Type u_1} :
@[simp]
theorem set.sup_eq_union {α : Type u_1} :
@[simp]
theorem set.inf_eq_inter {α : Type u_1} :
@[simp]
theorem set.le_eq_subset {α : Type u_1} :

set.lt_eq_ssubset is defined further down

@[simp]
theorem set.compl_eq_compl {α : Type u_1} :
@[instance]
def set.has_coe_to_sort {α : Type u_1} :

Coercion from a set to the corresponding subtype.

Equations
@[instance]
def set.pi_set_coe.can_lift (ι : Type u) (α : ι → Type v) [ne : ∀ (i : ι), nonempty (α i)] (s : set ι) :
can_lift (Π (i : s), α i) (Π (i : ι), α i)
Equations
@[instance]
def set.pi_set_coe.can_lift' (ι : Type u) (α : Type v) [ne : nonempty α] (s : set ι) :
can_lift (s → α) (ι → α)
Equations
@[instance]
def set.set_coe.can_lift {α : Type u_1} (s : set α) :
Equations
theorem set.set_coe_eq_subtype {α : Type u} (s : set α) :
s = {x // x s}
@[simp]
theorem set_coe.forall {α : Type u} {s : set α} {p : s → Prop} :
(∀ (x : s), p x) ∀ (x : α) (h : x s), p x, h⟩
@[simp]
theorem set_coe.exists {α : Type u} {s : set α} {p : s → Prop} :
(∃ (x : s), p x) ∃ (x : α) (h : x s), p x, h⟩
theorem set_coe.exists' {α : Type u} {s : set α} {p : Π (x : α), x s → Prop} :
(∃ (x : α) (h : x s), p x h) ∃ (x : s), p x _
theorem set_coe.forall' {α : Type u} {s : set α} {p : Π (x : α), x s → Prop} :
(∀ (x : α) (h : x s), p x h) ∀ (x : s), p x _
@[simp]
theorem set_coe_cast {α : Type u} {s t : set α} (H' : s = t) (H : s = t) (x : s) :
cast H x = x.val, _⟩
theorem set_coe.ext {α : Type u} {s : set α} {a b : s} :
a = ba = b
theorem set_coe.ext_iff {α : Type u} {s : set α} {a b : s} :
a = b a = b
theorem subtype.mem {α : Type u_1} {s : set α} (p : s) :
p s

See also subtype.prop

theorem eq.subset {α : Type u_1} {s t : set α} :
s = ts t
@[instance]
def set.inhabited {α : Type u} :
Equations
@[ext]
theorem set.ext {α : Type u} {a b : set α} (h : ∀ (x : α), x a x b) :
a = b
theorem set.ext_iff {α : Type u} {s t : set α} :
s = t ∀ (x : α), x s x t
theorem set.mem_of_mem_of_subset {α : Type u} {x : α} {s t : set α} (hx : x s) (h : s t) :
x t

Lemmas about mem and set_of #

@[simp]
theorem set.mem_set_of_eq {α : Type u} {a : α} {p : α → Prop} :
a {a : α | p a} = p a
theorem set.nmem_set_of_eq {α : Type u} {a : α} {P : α → Prop} :
a {a : α | P a} = ¬P a
@[simp]
theorem set.set_of_mem_eq {α : Type u} {s : set α} :
{x : α | x s} = s
theorem set.set_of_set {α : Type u} {s : set α} :
set_of s = s
theorem set.set_of_app_iff {α : Type u} {p : α → Prop} {x : α} :
{x : α | p x} x p x
theorem set.mem_def {α : Type u} {a : α} {s : set α} :
a s s a
@[instance]
def set.decidable_set_of {α : Type u} (p : α → Prop) [H : decidable_pred p] :
decidable_pred (λ (_x : α), _x {a : α | p a})
Equations
@[simp]
theorem set.set_of_subset_set_of {α : Type u} {p q : α → Prop} :
{a : α | p a} {a : α | q a} ∀ (a : α), p aq a
@[simp]
theorem set.sep_set_of {α : Type u} {p q : α → Prop} :
{a ∈ {a : α | p a} | q a} = {a : α | p a q a}
theorem set.set_of_and {α : Type u} {p q : α → Prop} :
{a : α | p a q a} = {a : α | p a} {a : α | q a}
theorem set.set_of_or {α : Type u} {p q : α → Prop} :
{a : α | p a q a} = {a : α | p a} {a : α | q a}

Lemmas about subsets #

theorem set.subset_def {α : Type u} {s t : set α} :
s t = ∀ (x : α), x sx t
theorem set.subset.refl {α : Type u} (a : set α) :
a a
theorem set.subset.rfl {α : Type u} {s : set α} :
s s
theorem set.subset.trans {α : Type u} {a b c : set α} (ab : a b) (bc : b c) :
a c
theorem set.mem_of_eq_of_mem {α : Type u} {x y : α} {s : set α} (hx : x = y) (h : y s) :
x s
theorem set.subset.antisymm {α : Type u} {a b : set α} (h₁ : a b) (h₂ : b a) :
a = b
theorem set.subset.antisymm_iff {α : Type u} {a b : set α} :
a = b a b b a
theorem set.eq_of_subset_of_subset {α : Type u} {a b : set α} :
a bb aa = b
theorem set.mem_of_subset_of_mem {α : Type u} {s₁ s₂ : set α} {a : α} (h : s₁ s₂) :
a s₁a s₂
theorem set.not_mem_subset {α : Type u} {a : α} {s t : set α} (h : s t) :
a ta s
theorem set.not_subset {α : Type u} {s t : set α} :
¬s t ∃ (a : α) (H : a s), a t

Definition of strict subsets s ⊂ t and basic properties. #

@[instance]
def set.has_ssubset {α : Type u} :
Equations
@[simp]
theorem set.ssubset_def {α : Type u} {s t : set α} :
s t = (s t ¬t s)
theorem set.eq_or_ssubset_of_subset {α : Type u} {s t : set α} (h : s t) :
s = t s t
theorem set.exists_of_ssubset {α : Type u} {s t : set α} (h : s t) :
∃ (x : α) (H : x t), x s
theorem set.ssubset_iff_subset_ne {α : Type u} {s t : set α} :
s t s t s t
theorem set.ssubset_iff_of_subset {α : Type u} {s t : set α} (h : s t) :
s t ∃ (x : α) (H : x t), x s
theorem set.ssubset_of_ssubset_of_subset {α : Type u} {s₁ s₂ s₃ : set α} (hs₁s₂ : s₁ s₂) (hs₂s₃ : s₂ s₃) :
s₁ s₃
theorem set.ssubset_of_subset_of_ssubset {α : Type u} {s₁ s₂ s₃ : set α} (hs₁s₂ : s₁ s₂) (hs₂s₃ : s₂ s₃) :
s₁ s₃
theorem set.not_mem_empty {α : Type u} (x : α) :
@[simp]
theorem set.not_not_mem {α : Type u} {a : α} {s : set α} :
¬a s a s

Non-empty sets #

def set.nonempty {α : Type u} (s : set α) :
Prop

The property s.nonempty expresses the fact that the set s is not empty. It should be used in theorem assumptions instead of ∃ x, x ∈ s or s ≠ ∅ as it gives access to a nice API thanks to the dot notation.

Equations
@[simp]
theorem set.nonempty_coe_sort {α : Type u} (s : set α) :
theorem set.nonempty_def {α : Type u} {s : set α} :
s.nonempty ∃ (x : α), x s
theorem set.nonempty_of_mem {α : Type u} {s : set α} {x : α} (h : x s) :
theorem set.nonempty.not_subset_empty {α : Type u} {s : set α} :
theorem set.nonempty.ne_empty {α : Type u} {s : set α} :
s.nonemptys
@[simp]
theorem set.not_nonempty_empty {α : Type u} :
def set.nonempty.some {α : Type u} {s : set α} (h : s.nonempty) :
α

Extract a witness from s.nonempty. This function might be used instead of case analysis on the argument. Note that it makes a proof depend on the classical.choice axiom.

Equations
theorem set.nonempty.some_mem {α : Type u} {s : set α} (h : s.nonempty) :
h.some s
theorem set.nonempty.mono {α : Type u} {s t : set α} (ht : s t) (hs : s.nonempty) :
theorem set.nonempty_of_not_subset {α : Type u} {s t : set α} (h : ¬s t) :
(s \ t).nonempty
theorem set.nonempty_of_ssubset {α : Type u} {s t : set α} (ht : s t) :
(t \ s).nonempty
theorem set.nonempty.of_diff {α : Type u} {s t : set α} (h : (s \ t).nonempty) :
theorem set.nonempty_of_ssubset' {α : Type u} {s t : set α} (ht : s t) :
theorem set.nonempty.inl {α : Type u} {s t : set α} (hs : s.nonempty) :
theorem set.nonempty.inr {α : Type u} {s t : set α} (ht : t.nonempty) :
@[simp]
theorem set.union_nonempty {α : Type u} {s t : set α} :
theorem set.nonempty.left {α : Type u} {s t : set α} (h : (s t).nonempty) :
theorem set.nonempty.right {α : Type u} {s t : set α} (h : (s t).nonempty) :
theorem set.nonempty_inter_iff_exists_right {α : Type u} {s t : set α} :
(s t).nonempty ∃ (x : t), x s
theorem set.nonempty_inter_iff_exists_left {α : Type u} {s t : set α} :
(s t).nonempty ∃ (x : s), x t
@[simp]
theorem set.univ_nonempty {α : Type u} [h : nonempty α] :
theorem set.nonempty.to_subtype {α : Type u} {s : set α} (h : s.nonempty) :
@[instance]
def set.univ.nonempty {α : Type u} [nonempty α] :
@[simp]
theorem set.nonempty_insert {α : Type u} (a : α) (s : set α) :
theorem set.nonempty_of_nonempty_subtype {α : Type u} {s : set α} [nonempty s] :

Lemmas about the empty set #

theorem set.empty_def {α : Type u} :
= {x : α | false}
@[simp]
theorem set.mem_empty_eq {α : Type u} (x : α) :
@[simp]
theorem set.set_of_false {α : Type u} :
{a : α | false} =
@[simp]
theorem set.empty_subset {α : Type u} (s : set α) :
theorem set.subset_empty_iff {α : Type u} {s : set α} :
theorem set.eq_empty_iff_forall_not_mem {α : Type u} {s : set α} :
s = ∀ (x : α), x s
theorem set.eq_empty_of_subset_empty {α : Type u} {s : set α} :
s s =
theorem set.eq_empty_of_is_empty {α : Type u} [is_empty α] (s : set α) :
s =
def set.unique_empty {α : Type u} [is_empty α] :
unique (set α)

There is exactly one set of a type that is empty.

Equations
theorem set.not_nonempty_iff_eq_empty {α : Type u} {s : set α} :
theorem set.empty_not_nonempty {α : Type u} :
theorem set.ne_empty_iff_nonempty {α : Type u} {s : set α} :
theorem set.eq_empty_or_nonempty {α : Type u} (s : set α) :
theorem set.subset_eq_empty {α : Type u} {s t : set α} (h : t s) (e : s = ) :
t =
theorem set.ball_empty_iff {α : Type u} {p : α → Prop} :
(∀ (x : α), x p x) true
@[instance]

Universal set. #

In Lean @univ α (or univ : set α) is the set that contains all elements of type α. Mathematically it is the same as α but it has a different type.

@[simp]
theorem set.set_of_true {α : Type u} :
{x : α | true} = set.univ
@[simp]
theorem set.mem_univ {α : Type u} (x : α) :
@[simp]
theorem set.univ_eq_empty_iff {α : Type u} :
theorem set.empty_ne_univ {α : Type u} [nonempty α] :
@[simp]
theorem set.subset_univ {α : Type u} (s : set α) :
theorem set.univ_subset_iff {α : Type u} {s : set α} :
theorem set.eq_univ_of_univ_subset {α : Type u} {s : set α} :
theorem set.eq_univ_iff_forall {α : Type u} {s : set α} :
s = set.univ ∀ (x : α), x s
theorem set.eq_univ_of_forall {α : Type u} {s : set α} :
(∀ (x : α), x s)s = set.univ
theorem set.eq_univ_of_subset {α : Type u} {s t : set α} (h : s t) (hs : s = set.univ) :
theorem set.exists_mem_of_nonempty (α : Type u_1) [nonempty α] :
∃ (x : α), x set.univ
@[instance]
def set.univ_decidable {α : Type u} :
decidable_pred (λ (_x : α), _x set.univ)
Equations
theorem set.ne_univ_iff_exists_not_mem {α : Type u_1} (s : set α) :
s set.univ ∃ (a : α), a s
theorem set.not_subset_iff_exists_mem_not_mem {α : Type u_1} {s t : set α} :
¬s t ∃ (x : α), x s x t
def set.diagonal (α : Type u_1) :
set × α)

diagonal α is the subset of α × α consisting of all pairs of the form (a, a).

Equations
@[simp]
theorem set.mem_diagonal {α : Type u_1} (x : α) :
(x, x) set.diagonal α

Lemmas about union #

theorem set.union_def {α : Type u} {s₁ s₂ : set α} :
s₁ s₂ = {a : α | a s₁ a s₂}
theorem set.mem_union_left {α : Type u} {x : α} {a : set α} (b : set α) :
x ax a b
theorem set.mem_union_right {α : Type u} {x : α} {b : set α} (a : set α) :
x bx a b
theorem set.mem_or_mem_of_mem_union {α : Type u} {x : α} {a b : set α} (H : x a b) :
x a x b
theorem set.mem_union.elim {α : Type u} {x : α} {a b : set α} {P : Prop} (H₁ : x a b) (H₂ : x a → P) (H₃ : x b → P) :
P
theorem set.mem_union {α : Type u} (x : α) (a b : set α) :
x a b x a x b
@[simp]
theorem set.mem_union_eq {α : Type u} (x : α) (a b : set α) :
x a b = (x a x b)
@[simp]
theorem set.union_self {α : Type u} (a : set α) :
a a = a
@[simp]
theorem set.union_empty {α : Type u} (a : set α) :
a = a
@[simp]
theorem set.empty_union {α : Type u} (a : set α) :
a = a
theorem set.union_comm {α : Type u} (a b : set α) :
a b = b a
theorem set.union_assoc {α : Type u} (a b c : set α) :
a b c = a (b c)
@[instance]
@[instance]
theorem set.union_left_comm {α : Type u} (s₁ s₂ s₃ : set α) :
s₁ (s₂ s₃) = s₂ (s₁ s₃)
theorem set.union_right_comm {α : Type u} (s₁ s₂ s₃ : set α) :
s₁ s₂ s₃ = s₁ s₃ s₂
@[simp]
theorem set.union_eq_left_iff_subset {α : Type u} {s t : set α} :
s t = s t s
@[simp]
theorem set.union_eq_right_iff_subset {α : Type u} {s t : set α} :
s t = t s t
theorem set.union_eq_self_of_subset_left {α : Type u} {s t : set α} (h : s t) :
s t = t
theorem set.union_eq_self_of_subset_right {α : Type u} {s t : set α} (h : t s) :
s t = s
@[simp]
theorem set.subset_union_left {α : Type u} (s t : set α) :
s s t
@[simp]
theorem set.subset_union_right {α : Type u} (s t : set α) :
t s t
theorem set.union_subset {α : Type u} {s t r : set α} (sr : s r) (tr : t r) :
s t r
@[simp]
theorem set.union_subset_iff {α : Type u} {s t u : set α} :
s t u s u t u
theorem set.union_subset_union {α : Type u} {s₁ s₂ t₁ t₂ : set α} (h₁ : s₁ s₂) (h₂ : t₁ t₂) :
s₁ t₁ s₂ t₂
theorem set.union_subset_union_left {α : Type u} {s₁ s₂ : set α} (t : set α) (h : s₁ s₂) :
s₁ t s₂ t
theorem set.union_subset_union_right {α : Type u} (s : set α) {t₁ t₂ : set α} (h : t₁ t₂) :
s t₁ s t₂
theorem set.subset_union_of_subset_left {α : Type u} {s t : set α} (h : s t) (u : set α) :
s t u
theorem set.subset_union_of_subset_right {α : Type u} {s u : set α} (h : s u) (t : set α) :
s t u
@[simp]
theorem set.union_empty_iff {α : Type u} {s t : set α} :
s t = s = t =
@[simp]
theorem set.union_univ {α : Type u} {s : set α} :
@[simp]
theorem set.univ_union {α : Type u} {s : set α} :

Lemmas about intersection #

theorem set.inter_def {α : Type u} {s₁ s₂ : set α} :
s₁ s₂ = {a : α | a s₁ a s₂}
theorem set.mem_inter_iff {α : Type u} (x : α) (a b : set α) :
x a b x a x b
@[simp]
theorem set.mem_inter_eq {α : Type u} (x : α) (a b : set α) :
x a b = (x a x b)
theorem set.mem_inter {α : Type u} {x : α} {a b : set α} (ha : x a) (hb : x b) :
x a b
theorem set.mem_of_mem_inter_left {α : Type u} {x : α} {a b : set α} (h : x a b) :
x a
theorem set.mem_of_mem_inter_right {α : Type u} {x : α} {a b : set α} (h : x a b) :
x b
@[simp]
theorem set.inter_self {α : Type u} (a : set α) :
a a = a
@[simp]
theorem set.inter_empty {α : Type u} (a : set α) :
@[simp]
theorem set.empty_inter {α : Type u} (a : set α) :
theorem set.inter_comm {α : Type u} (a b : set α) :
a b = b a
theorem set.inter_assoc {α : Type u} (a b c : set α) :
a b c = a (b c)
@[instance]
@[instance]
theorem set.inter_left_comm {α : Type u} (s₁ s₂ s₃ : set α) :
s₁ (s₂ s₃) = s₂ (s₁ s₃)
theorem set.inter_right_comm {α : Type u} (s₁ s₂ s₃ : set α) :
s₁ s₂ s₃ = s₁ s₃ s₂
@[simp]
theorem set.inter_subset_left {α : Type u} (s t : set α) :
s t s
@[simp]
theorem set.inter_subset_right {α : Type u} (s t : set α) :
s t t
theorem set.subset_inter {α : Type u} {s t r : set α} (rs : r s) (rt : r t) :
r s t
@[simp]
theorem set.subset_inter_iff {α : Type u} {s t r : set α} :
r s t r s r t
@[simp]
theorem set.inter_eq_left_iff_subset {α : Type u} {s t : set α} :
s t = s s t
@[simp]
theorem set.inter_eq_right_iff_subset {α : Type u} {s t : set α} :
s t = t t s
theorem set.inter_eq_self_of_subset_left {α : Type u} {s t : set α} :
s ts t = s
theorem set.inter_eq_self_of_subset_right {α : Type u} {s t : set α} :
t ss t = t
@[simp]
theorem set.inter_univ {α : Type u} (a : set α) :
@[simp]
theorem set.univ_inter {α : Type u} (a : set α) :
theorem set.inter_subset_inter {α : Type u} {s₁ s₂ t₁ t₂ : set α} (h₁ : s₁ t₁) (h₂ : s₂ t₂) :
s₁ s₂ t₁ t₂
theorem set.inter_subset_inter_left {α : Type u} {s t : set α} (u : set α) (H : s t) :
s u t u
theorem set.inter_subset_inter_right {α : Type u} {s t : set α} (u : set α) (H : s t) :
u s u t
theorem set.union_inter_cancel_left {α : Type u} {s t : set α} :
(s t) s = s
theorem set.union_inter_cancel_right {α : Type u} {s t : set α} :
(s t) t = t

Distributivity laws #

theorem set.inter_distrib_left {α : Type u} (s t u : set α) :
s (t u) = s t s u
theorem set.inter_union_distrib_left {α : Type u} {s t u : set α} :
s (t u) = s t s u
theorem set.inter_distrib_right {α : Type u} (s t u : set α) :
(s t) u = s u t u
theorem set.union_inter_distrib_right {α : Type u} {s t u : set α} :
(s t) u = s u t u
theorem set.union_distrib_left {α : Type u} (s t u : set α) :
s t u = (s t) (s u)
theorem set.union_inter_distrib_left {α : Type u} {s t u : set α} :
s t u = (s t) (s u)
theorem set.union_distrib_right {α : Type u} (s t u : set α) :
s t u = (s u) (t u)
theorem set.inter_union_distrib_right {α : Type u} {s t u : set α} :
s t u = (s u) (t u)

Lemmas about insert #

insert α s is the set {α} ∪ s.

theorem set.insert_def {α : Type u} (x : α) (s : set α) :
insert x s = {y : α | y = x y s}
@[simp]
theorem set.subset_insert {α : Type u} (x : α) (s : set α) :
s insert x s
theorem set.mem_insert {α : Type u} (x : α) (s : set α) :
x insert x s
theorem set.mem_insert_of_mem {α : Type u} {x : α} {s : set α} (y : α) :
x sx insert y s
theorem set.eq_or_mem_of_mem_insert {α : Type u} {x a : α} {s : set α} :
x insert a sx = a x s
theorem set.mem_of_mem_insert_of_ne {α : Type u} {x a : α} {s : set α} :
x insert a sx ax s
@[simp]
theorem set.mem_insert_iff {α : Type u} {x a : α} {s : set α} :
x insert a s x = a x s
@[simp]
theorem set.insert_eq_of_mem {α : Type u} {a : α} {s : set α} (h : a s) :
insert a s = s
theorem set.ne_insert_of_not_mem {α : Type u} {s : set α} (t : set α) {a : α} :
a ss insert a t
theorem set.insert_subset {α : Type u} {a : α} {s t : set α} :
insert a s t a t s t
theorem set.insert_subset_insert {α : Type u} {a : α} {s t : set α} (h : s t) :
insert a s insert a t
theorem set.insert_subset_insert_iff {α : Type u} {a : α} {s t : set α} (ha : a s) :
insert a s insert a t s t
theorem set.ssubset_iff_insert {α : Type u} {s t : set α} :
s t ∃ (a : α) (H : a s), insert a s t
theorem set.ssubset_insert {α : Type u} {s : set α} {a : α} (h : a s) :
s insert a s
theorem set.insert_comm {α : Type u} (a b : α) (s : set α) :
insert a (insert b s) = insert b (insert a s)
theorem set.insert_union {α : Type u} {a : α} {s t : set α} :
insert a s t = insert a (s t)
@[simp]
theorem set.union_insert {α : Type u} {a : α} {s t : set α} :
s insert a t = insert a (s t)
theorem set.insert_nonempty {α : Type u} (a : α) (s : set α) :
@[instance]
def set.has_insert.insert.nonempty {α : Type u} (a : α) (s : set α) :
theorem set.insert_inter {α : Type u} (x : α) (s t : set α) :
insert x (s t) = insert x s insert x t
theorem set.forall_of_forall_insert {α : Type u} {P : α → Prop} {a : α} {s : set α} (H : ∀ (x : α), x insert a sP x) (x : α) (h : x s) :
P x
theorem set.forall_insert_of_forall {α : Type u} {P : α → Prop} {a : α} {s : set α} (H : ∀ (x : α), x sP x) (ha : P a) (x : α) (h : x insert a s) :
P x
theorem set.bex_insert_iff {α : Type u} {P : α → Prop} {a : α} {s : set α} :
(∃ (x : α) (H : x insert a s), P x) P a ∃ (x : α) (H : x s), P x
theorem set.ball_insert_iff {α : Type u} {P : α → Prop} {a : α} {s : set α} :
(∀ (x : α), x insert a sP x) P a ∀ (x : α), x sP x

Lemmas about singletons #

theorem set.singleton_def {α : Type u} (a : α) :
{a} = {a}
@[simp]
theorem set.mem_singleton_iff {α : Type u} {a b : α} :
a {b} a = b
@[simp]
theorem set.set_of_eq_eq_singleton {α : Type u} {a : α} :
{n : α | n = a} = {a}
@[simp]
theorem set.set_of_eq_eq_singleton' {α : Type u} {a : α} :
{x : α | a = x} = {a}
@[simp]
theorem set.mem_singleton {α : Type u} (a : α) :
a {a}
theorem set.eq_of_mem_singleton {α : Type u} {x y : α} (h : x {y}) :
x = y
@[simp]
theorem set.singleton_eq_singleton_iff {α : Type u} {x y : α} :
{x} = {y} x = y
theorem set.mem_singleton_of_eq {α : Type u} {x y : α} (H : x = y) :
x {y}
theorem set.insert_eq {α : Type u} (x : α) (s : set α) :
insert x s = {x} s
@[simp]
theorem set.pair_eq_singleton {α : Type u} (a : α) :
{a, a} = {a}
theorem set.pair_comm {α : Type u} (a b : α) :
{a, b} = {b, a}
@[simp]
theorem set.singleton_nonempty {α : Type u} (a : α) :
@[simp]
theorem set.singleton_subset_iff {α : Type u} {a : α} {s : set α} :
{a} s a s
theorem set.set_compr_eq_eq_singleton {α : Type u} {a : α} :
{b : α | b = a} = {a}
@[simp]
theorem set.singleton_union {α : Type u} {a : α} {s : set α} :
{a} s = insert a s
@[simp]
theorem set.union_singleton {α : Type u} {a : α} {s : set α} :
s {a} = insert a s
@[simp]
theorem set.singleton_inter_nonempty {α : Type u} {a : α} {s : set α} :
({a} s).nonempty a s
@[simp]
theorem set.inter_singleton_nonempty {α : Type u} {a : α} {s : set α} :
(s {a}).nonempty a s
@[simp]
theorem set.singleton_inter_eq_empty {α : Type u} {a : α} {s : set α} :
{a} s = a s
@[simp]
theorem set.inter_singleton_eq_empty {α : Type u} {a : α} {s : set α} :
s {a} = a s
theorem set.nmem_singleton_empty {α : Type u} {s : set α} :
@[instance]
def set.unique_singleton {α : Type u} (a : α) :
Equations
theorem set.eq_singleton_iff_unique_mem {α : Type u} {s : set α} {a : α} :
s = {a} a s ∀ (x : α), x sx = a
theorem set.eq_singleton_iff_nonempty_unique_mem {α : Type u} {s : set α} {a : α} :
s = {a} s.nonempty ∀ (x : α), x sx = a
@[simp]
theorem set.default_coe_singleton {α : Type u} (x : α) :
default {x} = x, _⟩

Lemmas about sets defined as {x ∈ s | p x}. #

theorem set.mem_sep {α : Type u} {s : set α} {p : α → Prop} {x : α} (xs : x s) (px : p x) :
x {x ∈ s | p x}
@[simp]
theorem set.sep_mem_eq {α : Type u} {s t : set α} :
{x ∈ s | x t} = s t
@[simp]
theorem set.mem_sep_eq {α : Type u} {s : set α} {p : α → Prop} {x : α} :
x {x ∈ s | p x} = (x s p x)
theorem set.mem_sep_iff {α : Type u} {s : set α} {p : α → Prop} {x : α} :
x {x ∈ s | p x} x s p x
theorem set.eq_sep_of_subset {α : Type u} {s t : set α} (h : s t) :
s = {x ∈ t | x s}
@[simp]
theorem set.sep_subset {α : Type u} (s : set α) (p : α → Prop) :
{x ∈ s | p x} s
theorem set.forall_not_of_sep_empty {α : Type u} {s : set α} {p : α → Prop} (H : {x ∈ s | p x} = ) (x : α) :
x s¬p x
@[simp]
theorem set.sep_univ {α : Type u_1} {p : α → Prop} :
{a ∈ set.univ | p a} = {a : α | p a}
@[simp]
theorem set.sep_true {α : Type u} {s : set α} :
{a ∈ s | true} = s
@[simp]
theorem set.sep_false {α : Type u} {s : set α} :
{a ∈ s | false} =
theorem set.sep_inter_sep {α : Type u} {s : set α} {p q : α → Prop} :
{x ∈ s | p x} {x ∈ s | q x} = {x ∈ s | p x q x}
@[simp]
theorem set.subset_singleton_iff {α : Type u_1} {s : set α} {x : α} :
s {x} ∀ (y : α), y sy = x
theorem set.subset_singleton_iff_eq {α : Type u} {s : set α} {x : α} :
s {x} s = s = {x}
theorem set.ssubset_singleton_iff {α : Type u} {s : set α} {x : α} :
s {x} s =
theorem set.eq_empty_of_ssubset_singleton {α : Type u} {s : set α} {x : α} (hs : s {x}) :
s =

Lemmas about complement #

theorem set.mem_compl {α : Type u} {s : set α} {x : α} (h : x s) :
x s
theorem set.compl_set_of {α : Type u_1} (p : α → Prop) :
{a : α | p a} = {a : α | ¬p a}
theorem set.not_mem_of_mem_compl {α : Type u} {s : set α} {x : α} (h : x s) :
x s
@[simp]
theorem set.mem_compl_eq {α : Type u} (s : set α) (x : α) :
x s = (x s)
theorem set.mem_compl_iff {α : Type u} (s : set α) (x : α) :
x s x s
@[simp]
theorem set.inter_compl_self {α : Type u} (s : set α) :
@[simp]
theorem set.compl_inter_self {α : Type u} (s : set α) :
@[simp]
theorem set.compl_empty {α : Type u} :
@[simp]
theorem set.compl_union {α : Type u} (s t : set α) :
(s t) = s t
theorem set.compl_inter {α : Type u} (s t : set α) :
(s t) = s t
@[simp]
theorem set.compl_univ {α : Type u} :
@[simp]
theorem set.compl_empty_iff {α : Type u} {s : set α} :
@[simp]
theorem set.compl_univ_iff {α : Type u} {s : set α} :
theorem set.nonempty_compl {α : Type u} {s : set α} :
theorem set.mem_compl_singleton_iff {α : Type u} {a x : α} :
x {a} x a
theorem set.compl_singleton_eq {α : Type u} (a : α) :
{a} = {x : α | x a}
@[simp]
theorem set.compl_ne_eq_singleton {α : Type u} (a : α) :
{x : α | x a} = {a}
theorem set.union_eq_compl_compl_inter_compl {α : Type u} (s t : set α) :
s t = (s t)
theorem set.inter_eq_compl_compl_union_compl {α : Type u} (s t : set α) :
s t = (s t)
@[simp]
theorem set.union_compl_self {α : Type u} (s : set α) :
@[simp]
theorem set.compl_union_self {α : Type u} (s : set α) :
theorem set.compl_comp_compl {α : Type u} :
theorem set.compl_subset_comm {α : Type u} {s t : set α} :
s t t s
@[simp]
theorem set.compl_subset_compl {α : Type u} {s t : set α} :
s t t s
theorem set.compl_subset_iff_union {α : Type u} {s t : set α} :
theorem set.subset_compl_comm {α : Type u} {s t : set α} :
s t t s
theorem set.subset_compl_iff_disjoint {α : Type u} {s t : set α} :
s t s t =
theorem set.subset_compl_singleton_iff {α : Type u} {a : α} {s : set α} :
s {a} a s
theorem set.inter_subset {α : Type u} (a b c : set α) :
a b c a b c
theorem set.inter_compl_nonempty_iff {α : Type u} {s t : set α} :

Lemmas about set difference #

theorem set.diff_eq {α : Type u} (s t : set α) :
s \ t = s t
@[simp]
theorem set.mem_diff {α : Type u} {s t : set α} (x : α) :
x s \ t x s x t
theorem set.mem_diff_of_mem {α : Type u} {s t : set α} {x : α} (h1 : x s) (h2 : x t) :
x s \ t
theorem set.mem_of_mem_diff {α : Type u} {s t : set α} {x : α} (h : x s \ t) :
x s
theorem set.not_mem_of_mem_diff {α : Type u} {s t : set α} {x : α} (h : x s \ t) :
x t
theorem set.diff_eq_compl_inter {α : Type u} {s t : set α} :
s \ t = t s
theorem set.nonempty_diff {α : Type u} {s t : set α} :
(s \ t).nonempty ¬s t
theorem set.diff_subset {α : Type u} (s t : set α) :
s \ t s
theorem set.union_diff_cancel' {α : Type u} {s t u : set α} (h₁ : s t) (h₂ : t u) :
t u \ s = u
theorem set.union_diff_cancel {α : Type u} {s t : set α} (h : s t) :
s t \ s = t
theorem set.union_diff_cancel_left {α : Type u} {s t : set α} (h : s t ) :
(s t) \ s = t
theorem set.union_diff_cancel_right {α : Type u} {s t : set α} (h : s t ) :
(s t) \ t = s
@[simp]
theorem set.union_diff_left {α : Type u} {s t : set α} :
(s t) \ s = t \ s
@[simp]
theorem set.union_diff_right {α : Type u} {s t : set α} :
(s t) \ t = s \ t
theorem set.union_diff_distrib {α : Type u} {s t u : set α} :
(s t) \ u = s \ u t \ u
theorem set.inter_diff_assoc {α : Type u} (a b c : set α) :
a b \ c = a (b \ c)
@[simp]
theorem set.inter_diff_self {α : Type u} (a b : set α) :
a (b \ a) =
@[simp]
theorem set.inter_union_diff {α : Type u} (s t : set α) :
s t s \ t = s
@[simp]
theorem set.diff_union_inter {α : Type u} (s t : set α) :
s \ t s t = s
@[simp]
theorem set.inter_union_compl {α : Type u} (s t : set α) :
s t s t = s
theorem set.diff_subset_diff {α : Type u} {s₁ s₂ t₁ t₂ : set α} :
s₁ s₂t₂ t₁s₁ \ t₁ s₂ \ t₂
theorem set.diff_subset_diff_left {α : Type u} {s₁ s₂ t : set α} (h : s₁ s₂) :
s₁ \ t s₂ \ t
theorem set.diff_subset_diff_right {α : Type u} {s t u : set α} (h : t u) :
s \ u s \ t
theorem set.compl_eq_univ_diff {α : Type u} (s : set α) :
@[simp]
theorem set.empty_diff {α : Type u} (s : set α) :
theorem set.diff_eq_empty {α : Type u} {s t : set α} :
s \ t = s t
@[simp]
theorem set.diff_empty {α : Type u} {s : set α} :
s \ = s
@[simp]
theorem set.diff_univ {α : Type u} (s : set α) :
theorem set.diff_diff {α : Type u} {s t u : set α} :
s \ t \ u = s \ (t u)
theorem set.diff_diff_comm {α : Type u} {s t u : set α} :
s \ t \ u = s \ u \ t
theorem set.diff_subset_iff {α : Type u} {s t u : set α} :
s \ t u s t u
theorem set.subset_diff_union {α : Type u} (s t : set α) :
s s \ t t
theorem set.diff_union_of_subset {α : Type u} {s t : set α} (h : t s) :
s \ t t = s
@[simp]
theorem set.diff_singleton_subset_iff {α : Type u} {x : α} {s t : set α} :
s \ {x} t s insert x t
theorem set.subset_diff_singleton {α : Type u} {x : α} {s t : set α} (h : s t) (hx : x s) :
s t \ {x}
theorem set.subset_insert_diff_singleton {α : Type u} (x : α) (s : set α) :
s insert x (s \ {x})
theorem set.diff_subset_comm {α : Type u} {s t u : set α} :
s \ t u s \ u t
theorem set.diff_inter {α : Type u} {s t u : set α} :
s \ (t u) = s \ t s \ u
theorem set.diff_inter_diff {α : Type u} {s t u : set α} :
s \ t (s \ u) = s \ (t u)
theorem set.diff_compl {α : Type u} {s t : set α} :
s \ t = s t
theorem set.diff_diff_right {α : Type u} {s t u : set α} :
s \ (t \ u) = s \ t s u
@[simp]
theorem set.insert_diff_of_mem {α : Type u} {a : α} {t : set α} (s : set α) (h : a t) :
insert a s \ t = s \ t
theorem set.insert_diff_of_not_mem {α : Type u} {a : α} {t : set α} (s : set α) (h : a t) :
insert a s \ t = insert a (s \ t)
theorem set.insert_diff_self_of_not_mem {α : Type u} {a : α} {s : set α} (h : a s) :
insert a s \ {a} = s
theorem set.insert_inter_of_mem {α : Type u} {s₁ s₂ : set α} {a : α} (h : a s₂) :
insert a s₁ s₂ = insert a (s₁ s₂)
theorem set.insert_inter_of_not_mem {α : Type u} {s₁ s₂ : set α} {a : α} (h : a s₂) :
insert a s₁ s₂ = s₁ s₂
@[simp]
theorem set.union_diff_self {α : Type u} {s t : set α} :
s t \ s = s t
@[simp]
theorem set.diff_union_self {α : Type u} {s t : set α} :
s \ t t = s t
@[simp]
theorem set.diff_inter_self {α : Type u} {a b : set α} :
b \ a a =
@[simp]
theorem set.diff_inter_self_eq_diff {α : Type u} {s t : set α} :
s \ (t s) = s \ t
@[simp]
theorem set.diff_self_inter {α : Type u} {s t : set α} :
s \ (s t) = s \ t
@[simp]
theorem set.diff_eq_self {α : Type u} {s t : set α} :
s \ t = s t s
@[simp]
theorem set.diff_singleton_eq_self {α : Type u} {a : α} {s : set α} (h : a s) :
s \ {a} = s
@[simp]
theorem set.insert_diff_singleton {α : Type u} {a : α} {s : set α} :
insert a (s \ {a}) = insert a s
@[simp]
theorem set.diff_self {α : Type u} {s : set α} :
s \ s =
theorem set.diff_diff_cancel_left {α : Type u} {s t : set α} (h : s t) :
t \ (t \ s) = s
theorem set.mem_diff_singleton {α : Type u} {x y : α} {s : set α} :
x s \ {y} x s x y
theorem set.mem_diff_singleton_empty {α : Type u} {s : set α} {t : set (set α)} :
s t \ {} s t s.nonempty
theorem set.union_eq_diff_union_diff_union_inter {α : Type u} (s t : set α) :
s t = s \ t t \ s s t

Powerset #

theorem set.mem_powerset {α : Type u} {x s : set α} (h : x s) :
theorem set.subset_of_mem_powerset {α : Type u} {x s : set α} (h : x 𝒫s) :
x s
@[simp]
theorem set.mem_powerset_iff {α : Type u} (x s : set α) :
x 𝒫s x s
theorem set.powerset_inter {α : Type u} (s t : set α) :
@[simp]
theorem set.powerset_mono {α : Type u} {s t : set α} :
@[simp]
theorem set.powerset_nonempty {α : Type u} {s : set α} :
@[simp]
theorem set.powerset_empty {α : Type u} :
@[simp]
theorem set.powerset_univ {α : Type u} :

If-then-else for sets #

def set.ite {α : Type u} (t s s' : set α) :
set α

ite for sets: set.ite t s s' ∩ t = s ∩ t, set.ite t s s' ∩ tᶜ = s' ∩ tᶜ. Defined as s ∩ t ∪ s' \ t.

Equations
@[simp]
theorem set.ite_inter_self {α : Type u} (t s s' : set α) :
t.ite s s' t = s t
@[simp]
theorem set.ite_compl {α : Type u} (t s s' : set α) :
t.ite s s' = t.ite s' s
@[simp]
theorem set.ite_inter_compl_self {α : Type u} (t s s' : set α) :
t.ite s s' t = s' t
@[simp]
theorem set.ite_diff_self {α : Type u} (t s s' : set α) :
t.ite s s' \ t = s' \ t
@[simp]
theorem set.ite_same {α : Type u} (t s : set α) :
t.ite s s = s
@[simp]
theorem set.ite_left {α : Type u} (s t : set α) :
s.ite s t = s t
@[simp]
theorem set.ite_right {α : Type u} (s t : set α) :
s.ite t s = t s
@[simp]
theorem set.ite_empty {α : Type u} (s s' : set α) :
.ite s s' = s'
@[simp]
theorem set.ite_univ {α : Type u} (s s' : set α) :
set.univ.ite s s' = s
@[simp]
theorem set.ite_empty_left {α : Type u} (t s : set α) :
t.ite s = s \ t
@[simp]
theorem set.ite_empty_right {α : Type u} (t s : set α) :
t.ite s = s t
theorem set.ite_mono {α : Type u} (t : set α) {s₁ s₁' s₂ s₂' : set α} (h : s₁ s₂) (h' : s₁' s₂') :
t.ite s₁ s₁' t.ite s₂ s₂'
theorem set.ite_subset_union {α : Type u} (t s s' : set α) :
t.ite s s' s s'
theorem set.inter_subset_ite {α : Type u} (t s s' : set α) :
s s' t.ite s s'
theorem set.ite_inter_inter {α : Type u} (t s₁ s₂ s₁' s₂' : set α) :
t.ite (s₁ s₂) (s₁' s₂') = t.ite s₁ s₁' t.ite s₂ s₂'
theorem set.ite_inter {α : Type u} (t s₁ s₂ s : set α) :
t.ite (s₁ s) (s₂ s) = t.ite s₁ s₂ s
theorem set.ite_inter_of_inter_eq {α : Type u} (t : set α) {s₁ s₂ s : set α} (h : s₁ s = s₂ s) :
t.ite s₁ s₂ s = s₁ s
theorem set.subset_ite {α : Type u} {t s s' u : set α} :
u t.ite s s' u t s u \ t s'

Inverse image #

def set.preimage {α : Type u} {β : Type v} (f : α → β) (s : set β) :
set α

The preimage of s : set β by f : α → β, written f ⁻¹' s, is the set of x : α such that f x ∈ s.

Equations
@[simp]
theorem set.preimage_empty {α : Type u} {β : Type v} {f : α → β} :
@[simp]
theorem set.mem_preimage {α : Type u} {β : Type v} {f : α → β} {s : set β} {a : α} :
a f ⁻¹' s f a s
theorem set.preimage_congr {α : Type u} {β : Type v} {f g : α → β} {s : set β} (h : ∀ (x : α), f x = g x) :
f ⁻¹' s = g ⁻¹' s
theorem set.preimage_mono {α : Type u} {β : Type v} {f : α → β} {s t : set β} (h : s t) :
f ⁻¹' s f ⁻¹' t
@[simp]
theorem set.preimage_univ {α : Type u} {β : Type v} {f : α → β} :
theorem set.subset_preimage_univ {α : Type u} {β : Type v} {f : α → β} {s : set α} :
@[simp]
theorem set.preimage_inter {α : Type u} {β : Type v} {f : α → β} {s t : set β} :
f ⁻¹' (s t) = f ⁻¹' s f ⁻¹' t
@[simp]
theorem set.preimage_union {α : Type u} {β : Type v} {f : α → β} {s t : set β} :
f ⁻¹' (s t) = f ⁻¹' s f ⁻¹' t
@[simp]
theorem set.preimage_compl {α : Type u} {β : Type v} {f : α → β} {s : set β} :
@[simp]
theorem set.preimage_diff {α : Type u} {β : Type v} (f : α → β) (s t : set β) :
f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t
@[simp]
theorem set.preimage_ite {α : Type u} {β : Type v} (f : α → β) (s t₁ t₂ : set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂)
@[simp]
theorem set.preimage_set_of_eq {α : Type u} {β : Type v} {p : α → Prop} {f : β → α} :
f ⁻¹' {a : α | p a} = {a : β | p (f a)}
@[simp]
theorem set.preimage_id {α : Type u} {s : set α} :
id ⁻¹' s = s
@[simp]
theorem set.preimage_id' {α : Type u} {s : set α} :
(λ (x : α), x) ⁻¹' s = s
@[simp]
theorem set.preimage_const_of_mem {α : Type u} {β : Type v} {b : β} {s : set β} (h : b s) :
(λ (x : α), b) ⁻¹' s = set.univ
@[simp]
theorem set.preimage_const_of_not_mem {α : Type u} {β : Type v} {b : β} {s : set β} (h : b s) :
(λ (x : α), b) ⁻¹' s =
theorem set.preimage_const {α : Type u} {β : Type v} (b : β) (s : set β) [decidable (b s)] :
(λ (x : α), b) ⁻¹' s = ite (b s) set.univ
theorem set.preimage_comp {α : Type u} {β : Type v} {γ : Type w} {f : α → β} {g : β → γ} {s : set γ} :
g f ⁻¹' s = f ⁻¹' (g ⁻¹' s)
theorem set.preimage_preimage {α : Type u} {β : Type v} {γ : Type w} {g : β → γ} {f : α → β} {s : set γ} :
f ⁻¹' (g ⁻¹' s) = (λ (x : α), g (f x)) ⁻¹' s
theorem set.eq_preimage_subtype_val_iff {α : Type u} {p : α → Prop} {s : set (subtype p)} {t : set α} :
s = subtype.val ⁻¹' t ∀ (x : α) (h : p x), x, h⟩ s x t

Image of a set under a function #

theorem set.mem_image_iff_bex {α : Type u} {β : Type v} {f : α → β} {s : set α} {y : β} :
y f '' s ∃ (x : α) (_x : x s), f x = y
theorem set.mem_image_eq {α : Type u} {β : Type v} (f : α → β) (s : set α) (y : β) :
y f '' s = ∃ (x : α), x s f x = y
@[simp]
theorem set.mem_image {α : Type u} {β : Type v} (f : α → β) (s : set α) (y : β) :
y f '' s ∃ (x : α), x s f x = y
theorem set.image_eta {α : Type u} {β : Type v} {s : set α} (f : α → β) :
f '' s = (λ (x : α), f x) '' s
theorem set.mem_image_of_mem {α : Type u} {β : Type v} (f : α → β) {x : α} {a : set α} (h : x a) :
f x f '' a
theorem function.injective.mem_set_image {α : Type u} {β : Type v} {f : α → β} (hf : function.injective f) {s : set α} {a : α} :
f a f '' s a s
theorem set.ball_image_iff {α : Type u} {β : Type v} {f : α → β} {s : set α} {p : β → Prop} :
(∀ (y : β), y f '' sp y) ∀ (x : α), x sp (f x)
theorem set.ball_image_of_ball {α : Type u} {β : Type v} {f : α → β} {s : set α} {p : β → Prop} (h : ∀ (x : α), x sp (f x)) (y : β) (H : y f '' s) :
p y
theorem set.bex_image_iff {α : Type u} {β : Type v} {f : α → β} {s : set α} {p : β → Prop} :
(∃ (y : β) (H : y f '' s), p y) ∃ (x : α) (H : x s), p (f x)
theorem set.mem_image_elim {α : Type u} {β : Type v} {f : α → β} {s : set α} {C : β → Prop} (h : ∀ (x : α), x sC (f x)) {y : β} :
y f '' sC y
theorem set.mem_image_elim_on {α : Type u} {β : Type v} {f : α → β} {s : set α} {C : β → Prop} {y : β} (h_y : y f '' s) (h : ∀ (x : α), x sC (f x)) :
C y
theorem set.image_congr {α : Type u} {β : Type v} {f g : α → β} {s : set α} (h : ∀ (a : α), a sf a = g a) :
f '' s = g '' s
theorem set.image_congr' {α : Type u} {β : Type v} {f g : α → β} {s : set α} (h : ∀ (x : α), f x = g x) :
f '' s = g '' s

A common special case of image_congr

theorem set.image_comp {α : Type u} {β : Type v} {γ : Type w} (f : β → γ) (g : α → β) (a : set α) :
f g '' a = f '' (g '' a)
theorem set.image_image {α : Type u} {β : Type v} {γ : Type w} (g : β → γ) (f : α → β) (s : set α) :
g '' (f '' s) = (λ (x : α), g (f x)) '' s

A variant of image_comp, useful for rewriting

theorem set.image_subset {α : Type u} {β : Type v} {a b : set α} (f : α → β) (h : a b) :
f '' a f '' b

Image is monotone with respect to . See set.monotone_image for the statement in terms of .

theorem set.image_union {α : Type u} {β : Type v} (f : α → β) (s t : set α) :
f '' (s t) = f '' s f '' t
@[simp]
theorem set.image_empty {α : Type u} {β : Type v} (f : α → β) :
theorem set.image_inter_subset {α : Type u} {β : Type v} (f : α → β) (s t : set α) :
f '' (s t) f '' s f '' t
theorem set.image_inter_on {α : Type u} {β : Type v} {f : α → β} {s t : set α} (h : ∀ (x : α), x t∀ (y : α), y sf x = f yx = y) :
f '' s f '' t = f '' (s t)
theorem set.image_inter {α : Type u} {β : Type v} {f : α → β} {s t : set α} (H : function.injective f) :
f '' s f '' t = f '' (s t)
theorem set.image_univ_of_surjective {β : Type v} {ι : Type u_1} {f : ι → β} (H : function.surjective f) :
@[simp]
theorem set.image_singleton {α : Type u} {β : Type v} {f : α → β} {a : α} :
f '' {a} = {f a}
@[simp]
theorem set.nonempty.image_const {α : Type u} {β : Type v} {s : set α} (hs : s.nonempty) (a : β) :
(λ (_x : α), a) '' s = {a}
@[simp]
theorem set.image_eq_empty {α : Type u_1} {β : Type u_2} {f : α → β} {s : set α} :
f '' s = s =
theorem set.mem_compl_image {α : Type u} (t : set α) (S : set (set α)) :
@[simp]
theorem set.image_id' {α : Type u} (s : set α) :
(λ (x : α), x) '' s = s

A variant of image_id

theorem set.image_id {α : Type u} (s : set α) :
id '' s = s
theorem set.compl_compl_image {α : Type u} (S : set (set α)) :
compl '' (compl '' S) = S
theorem set.image_insert_eq {α : Type u} {β : Type v} {f : α → β} {a : α} {s : set α} :
f '' insert a s = insert (f a) (f '' s)
theorem set.image_pair {α : Type u} {β : Type v} (f : α → β) (a b : α) :
f '' {a, b} = {f a, f b}
theorem set.image_subset_preimage_of_inverse {α : Type u} {β : Type v} {f : α → β} {g : β → α} (I : function.left_inverse g f) (s : set α) :
f '' s g ⁻¹' s
theorem set.preimage_subset_image_of_inverse {α : Type u} {β : Type v} {f : α → β} {g : β → α} (I : function.left_inverse g f) (s : set β) :
f ⁻¹' s g '' s
theorem set.image_eq_preimage_of_inverse {α : Type u} {β : Type v} {f : α → β} {g : β → α} (h₁ : function.left_inverse g f) (h₂ : function.right_inverse g f) :
theorem set.mem_image_iff_of_inverse {α : Type u} {β : Type v} {f : α → β} {g : β → α} {b : β} {s : set α} (h₁ : function.left_inverse g f) (h₂ : function.right_inverse g f) :
b f '' s g b s
theorem set.image_compl_subset {α : Type u} {β : Type v} {f : α → β} {s : set α} (H : function.injective f) :
f '' s (f '' s)
theorem set.subset_image_compl {α : Type u} {β : Type v} {f : α → β} {s : set α} (H : function.surjective f) :
(f '' s) f '' s
theorem set.image_compl_eq {α : Type u} {β : Type v} {f : α → β} {s : set α} (H : function.bijective f) :
f '' s = (f '' s)
theorem set.subset_image_diff {α : Type u} {β : Type v} (f : α → β) (s t : set α) :
f '' s \ f '' t f '' (s \ t)
theorem set.image_diff {α : Type u} {β : Type v} {f : α → β} (hf : function.injective f) (s t : set α) :
f '' (s \ t) = f '' s \ f '' t
theorem set.nonempty.image {α : Type u} {β : Type v} (f : α → β) {s : set α} :
s.nonempty(f '' s).nonempty
theorem set.nonempty.of_image {α : Type u} {β : Type v} {f : α → β} {s : set α} :
(f '' s).nonempty → s.nonempty
@[simp]
theorem set.nonempty_image_iff {α : Type u} {β : Type v} {f : α → β} {s : set α} :
theorem set.nonempty.preimage {α : Type u} {β : Type v} {s : set β} (hs : s.nonempty) {f : α → β} (hf : function.surjective f) :
@[instance]
def set.image.nonempty {α : Type u} {β : Type v} (f : α → β) (s : set α) [nonempty s] :
@[simp]
theorem set.image_subset_iff {α : Type u} {β : Type v} {s : set α} {t : set β} {f : α → β} :
f '' s t s f ⁻¹' t

image and preimage are a Galois connection

theorem set.image_preimage_subset {α : Type u} {β : Type v} (f : α → β) (s : set β) :
f '' (f ⁻¹' s) s
theorem set.subset_preimage_image {α : Type u} {β : Type v} (f : α → β) (s : set α) :
s f ⁻¹' (f '' s)
theorem set.preimage_image_eq {α : Type u} {β : Type v} {f : α → β} (s : set α) (h : function.injective f) :
f ⁻¹' (f '' s) = s
theorem set.image_preimage_eq {α : Type u} {β : Type v} {f : α → β} (s : set β) (h : function.surjective f) :
f '' (f ⁻¹' s) = s
theorem set.preimage_eq_preimage {α : Type u} {β : Type v} {s t : set α} {f : β → α} (hf : function.surjective f) :
f ⁻¹' s = f ⁻¹' t s = t
theorem set.image_inter_preimage {α : Type u} {β : Type v} (f : α → β) (s : set α) (t : set β) :
f '' (s f ⁻¹' t) = f '' s t
theorem set.image_preimage_inter {α : Type u} {β : Type v} (f : α → β) (s : set α) (t : set β) :
f '' (f ⁻¹' t s) = t f '' s
@[simp]
theorem set.image_inter_nonempty_iff {α : Type u} {β : Type v} {f : α → β} {s : set α} {t : set β} :
(f '' s t).nonempty (s f ⁻¹' t).nonempty
theorem set.image_diff_preimage {α : Type u} {β : Type v} {f : α → β} {s : set α} {t : set β} :
f '' (s \ f ⁻¹' t) = f '' s \ t
theorem set.compl_image_set_of {α : Type u} {p : set α → Prop} :
compl '' {s : set α | p s} = {s : set α | p s}
theorem set.inter_preimage_subset {α : Type u} {β : Type v} (s : set α) (t : set β) (f : α → β) :
s f ⁻¹' t f ⁻¹' (f '' s t)
theorem set.union_preimage_subset {α : Type u} {β : Type v} (s : set α) (t : set β) (f : α → β) :
s f ⁻¹' t f ⁻¹' (f '' s t)
theorem set.subset_image_union {α : Type u} {β : Type v} (f : α → β) (s : set α) (t : set β) :
f '' (s f ⁻¹' t) f '' s t
theorem set.preimage_subset_iff {α : Type u} {β : Type v} {A : set α} {B : set β} {f : α → β} :
f ⁻¹' B A ∀ (a : α), f a Ba A
theorem set.image_eq_image {α : Type u} {β : Type v} {s t : set α} {f : α → β} (hf : function.injective f) :
f '' s = f '' t s = t
theorem set.image_subset_image_iff {α : Type u} {β : Type v} {s t : set α} {f : α → β} (hf : function.injective f) :
f '' s f '' t s t
theorem set.prod_quotient_preimage_eq_image {α : Type u} {β : Type v} [s : setoid α] (g : quotient s → β) {h : α → β} (Hh : h = g quotient.mk) (r : set × β)) :
{x : quotient s × quotient s | (g x.fst, g x.snd) r} = (λ (a : α × α), (a.fst, a.snd)) '' ((λ (a : α × α), (h a.fst, h a.snd)) ⁻¹' r)
def set.image_factorization {α : Type u} {β : Type v} (f : α → β) (s : set α) :
s → (f '' s)

Restriction of f to s factors through s.image_factorization f : s → f '' s.

Equations
theorem set.image_factorization_eq {α : Type u} {β : Type v} {f : α → β} {s : set α} :
theorem set.surjective_onto_image {α : Type u} {β : Type v} {f : α → β} {s : set α} :

Subsingleton #

def set.subsingleton {α : Type u} (s : set α) :
Prop

A set s is a subsingleton, if it has at most one element.

Equations
theorem set.subsingleton.mono {α : Type u} {s t : set α} (ht : t.subsingleton) (hst : s t) :
theorem set.subsingleton.image {α : Type u} {β : Type v} {s : set α} (hs : s.subsingleton) (f : α → β) :
theorem set.subsingleton.eq_singleton_of_mem {α : Type u} {s : set α} (hs : s.subsingleton) {x : α} (hx : x s) :
s = {x}
@[simp]
theorem set.subsingleton_empty {α : Type u} :
@[simp]
theorem set.subsingleton_singleton {α : Type u} {a : α} :
theorem set.subsingleton_iff_singleton {α : Type u} {s : set α} {x : α} (hx : x s) :
theorem set.subsingleton.eq_empty_or_singleton {α : Type u} {s : set α} (hs : s.subsingleton) :
s = ∃ (x : α), s = {x}
theorem set.subsingleton.induction_on {α : Type u} {s : set α} {p : set α → Prop} (hs : s.subsingleton) (he : p ) (h₁ : ∀ (x : α), p {x}) :
p s
theorem set.subsingleton_of_subsingleton {α : Type u} [subsingleton α] {s : set α} :
theorem set.subsingleton_is_top (α : Type u_1) [partial_order α] :
{x : α | is_top x}.subsingleton
theorem set.subsingleton_is_bot (α : Type u_1) [partial_order α] :
{x : α | is_bot x}.subsingleton
@[simp]
theorem set.subsingleton_coe {α : Type u} (s : set α) :

s, coerced to a type, is a subsingleton type if and only if s is a subsingleton set.

theorem set.subsingleton.preimage {α : Type u} {β : Type v} {s : set β} (hs : s.subsingleton) {f : α → β} (hf : function.injective f) :

The preimage of a subsingleton under an injective map is a subsingleton.

theorem set.subsingleton_of_image {α : Type u_1} {β : Type u_2} {f : α → β} (hf : function.injective f) (s : set α) (hs : (f '' s).subsingleton) :

s is a subsingleton, if its image of an injective function is.

Lemmas about range of a function. #

def set.range {α : Type u} {ι : Sort x} (f : ι → α) :
set α

Range of a function.

This function is more flexible than f '' univ, as the image requires that the domain is in Type and not an arbitrary Sort.

Equations
@[simp]
theorem set.mem_range {α : Type u} {ι : Sort x} {f : ι → α} {x : α} :
x set.range f ∃ (y : ι), f y = x
@[simp]
theorem set.mem_range_self {α : Type u} {ι : Sort x} {f : ι → α} (i : ι) :
theorem set.forall_range_iff {α : Type u} {ι : Sort x} {f : ι → α} {p : α → Prop} :
(∀ (a : α), a set.range fp a) ∀ (i : ι), p (f i)
theorem set.forall_subtype_range_iff {α : Type u} {ι : Sort x} {f : ι → α} {p : (set.range f) → Prop} :
(∀ (a : (set.range f)), p a) ∀ (i : ι), p f i, _⟩
theorem set.exists_range_iff {α : Type u} {ι : Sort x} {f : ι → α} {p : α → Prop} :
(∃ (a : α) (H : a set.range f), p a) ∃ (i : ι), p (f i)
theorem set.exists_range_iff' {α : Type u} {ι : Sort x} {f : ι → α} {p : α → Prop} :
(∃ (a : α), a set.range f p a) ∃ (i : ι), p (f i)
theorem set.exists_subtype_range_iff {α : Type u} {ι : Sort x} {f : ι → α} {p : (set.range f) → Prop} :
(∃ (a : (set.range f)), p a) ∃ (i : ι), p f i, _⟩
theorem set.range_iff_surjective {α : Type u} {ι : Sort x} {f : ι → α} :
theorem function.surjective.range_eq {α : Type u} {ι : Sort x} {f : ι → α} :

Alias of range_iff_surjective.

@[simp]
theorem set.range_id {α : Type u} :
@[simp]
theorem prod.range_fst {α : Type u} {β : Type v} [nonempty β] :
@[simp]
theorem prod.range_snd {α : Type u} {β : Type v} [nonempty α] :
@[simp]
theorem set.range_eval {ι : Type u_1} {α : ι → Type u_2} [∀ (i : ι), nonempty (α i)] (i : ι) :
@[simp]
@[simp]
theorem set.range_inl_inter_range_inr {α : Type u} {β : Type v} :
@[simp]
@[simp]
theorem set.range_inr_inter_range_inl {α : Type u} {β : Type v} :
@[simp]
theorem set.preimage_inl_range_inr {α : Type u} {β : Type v} :
@[simp]
theorem set.preimage_inr_range_inl {α : Type u} {β : Type v} :
@[simp]
theorem set.range_quot_mk {α : Type u} (r : α → α → Prop) :
set.range (quot.mk r) = set.univ
@[simp]
theorem set.image_univ {α : Type u} {β : Type v} {f : α → β} :
theorem set.image_subset_range {α : Type u} {β : Type v} (f : α → β) (s : set α) :
theorem set.mem_range_of_mem_image {α : Type u} {β : Type v} (f : α → β) (s : set α) {x : β} (h : x f '' s) :
theorem set.range_comp {α : Type u} {β : Type v} {ι : Sort x} (g : α → β) (f : ι → α) :
theorem set.range_subset_iff {α : Type u} {ι : Sort x} {s : set α} {f : ι → α} :
set.range f s ∀ (y : ι), f y s
theorem set.range_comp_subset_range {α : Type u} {β : Type v} {γ : Type w} (f : α → β) (g : β → γ) :
theorem set.range_nonempty_iff_nonempty {α : Type u} {ι : Sort x} {f : ι → α} :
theorem set.range_nonempty {α : Type u} {ι : Sort x} [h : nonempty ι] (f : ι → α) :
@[simp]
theorem set.range_eq_empty_iff {α : Type u} {ι : Sort x} {f : ι → α} :
theorem set.range_eq_empty {α : Type u} {ι : Sort x} [is_empty ι] (f : ι → α) :
@[instance]
def set.range.nonempty {α : Type u} {ι : Sort x} [nonempty ι] (f : ι → α) :
@[simp]
theorem set.image_union_image_compl_eq_range {α : Type u} {β : Type v} {s : set α} (f : α → β) :
f '' s f '' s = set.range f
theorem set.image_preimage_eq_inter_range {α : Type u} {β : Type v} {f : α → β} {t : set β} :
f '' (f ⁻¹' t) = t set.range f
theorem set.image_preimage_eq_of_subset {α : Type u} {β : Type v} {f : α → β} {s : set β} (hs : s set.range f) :
f '' (f ⁻¹' s) = s
@[instance]
def set.set.can_lift {α : Type u} {β : Type v} [can_lift α β] :
can_lift (set α) (set β)
Equations
theorem set.image_preimage_eq_iff {α : Type u} {β : Type v} {f : α → β} {s : set β} :
f '' (f ⁻¹' s) = s s set.range f
theorem set.preimage_subset_preimage_iff {α : Type u} {β : Type v} {s t : set α} {f : β → α} (hs : s set.range f) :
f ⁻¹' s f ⁻¹' t s t
theorem set.preimage_eq_preimage' {α : Type u} {β : Type v} {s t : set α} {f : β → α} (hs : s set.range f) (ht : t set.range f) :
f ⁻¹' s = f ⁻¹' t s = t
@[simp]
theorem set.preimage_inter_range {α : Type u} {β : Type v} {f : α → β} {s : set β} :
@[simp]
theorem set.preimage_range_inter {α : Type u} {β : Type v} {f : α → β} {s : set β} :
theorem set.preimage_image_preimage {α : Type u} {β : Type v} {f : α → β} {s : set β} :
f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s
@[simp]
theorem set.quot_mk_range_eq {α : Type u} [setoid α] :
set.range (λ (x : α), x) = set.univ
theorem set.range_const_subset {α : Type u} {ι : Sort x} {c : α} :
set.range (λ (x : ι), c) {c}
@[simp]
theorem set.range_const {α : Type u} {ι : Sort x} [nonempty ι] {c : α} :
set.range (λ (x : ι), c) = {c}
theorem set.diagonal_eq_range {α : Type u_1} :
set.diagonal α = set.range (λ (x : α), (x, x))
theorem set.preimage_singleton_nonempty {α : Type u} {β : Type v} {f : α → β} {y : β} :
theorem set.preimage_singleton_eq_empty {α : Type u} {β : Type v} {f : α → β} {y : β} :
theorem set.range_subset_singleton {α : Type u} {ι : Sort x} {f : ι → α} {x : α} :
theorem set.image_compl_preimage {α : Type u} {β : Type v} {f : α → β} {s : set β} :
f '' (f ⁻¹' s) = set.range f \ s
@[simp]
theorem set.range_sigma_mk {α : Type u} {β : α → Type u_1} (a : α) :
def set.range_factorization {β : Type v} {ι : Sort x} (f : ι → β) :
ι → (set.range f)

Any map f : ι → β factors through a map range_factorization f : ι → range f.

Equations
theorem set.range_factorization_eq {β : Type v} {ι : Sort x} {f : ι → β} :
@[simp]
theorem set.range_factorization_coe {β : Type v} {ι : Sort x} (f : ι → β) (a : ι) :
theorem set.surjective_onto_range {α : Type u} {ι : Sort x} {f : ι → α} :
theorem set.image_eq_range {α : Type u} {β : Type v} (f : α → β) (s : set α) :
f '' s = set.range (λ (x : s), f x)
@[simp]
theorem set.sum.elim_range {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → γ) (g : β → γ) :
theorem set.range_ite_subset' {α : Type u} {β : Type v} {p : Prop} [decidable p] {f g : α → β} :
theorem set.range_ite_subset {α : Type u} {β : Type v} {p : α → Prop} [decidable_pred p] {f g : α → β} :
set.range (λ (x : α), ite (p x) (f x) (g x)) set.range f set.range g
@[simp]
theorem set.preimage_range {α : Type u} {β : Type v} (f : α → β) :
theorem set.range_unique {α : Type u} {ι : Sort x} {f : ι → α} [h : unique ι] :
set.range f = {f (default ι)}

The range of a function from a unique type contains just the function applied to its single value.

theorem set.range_diff_image_subset {α : Type u} {β : Type v} (f : α → β) (s : set α) :
set.range f \ f '' s f '' s
theorem