Equivalences and sets

In this file we provide lemmas linking equivalences to sets.

Some notable definitions are:

• Equiv.ofInjective: an injective function is (noncomputably) equivalent to its range.
• Equiv.setCongr: two equal sets are equivalent as types.
• Equiv.Set.union: a disjoint union of sets is equivalent to their Sum.

This file is separate from Equiv/Basic such that we do not require the full lattice structure on sets before defining what an equivalence is.

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@[simp]

theorem Equiv.range_eq_univ {α : Type u_1} {β : Type u_2} (e : α ≃ β) : Set.range ↑e = Set.univ

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theorem Equiv.image_eq_preimage {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : Set α) : ↑e '' s = ↑(Equiv.symm e) ⁻¹' s

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theorem Set.mem_image_equiv {α : Type u_1} {β : Type u_2} {S : Set α} {f : α ≃ β} {x : β} : x ∈ ↑f '' S ↔ ↑(Equiv.symm f) x ∈ S

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theorem Set.image_equiv_eq_preimage_symm {α : Type u_1} {β : Type u_2} (S : Set α) (f : α ≃ β) : ↑f '' S = ↑(Equiv.symm f) ⁻¹' S

Alias for Equiv.image_eq_preimage

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theorem Set.preimage_equiv_eq_image_symm {α : Type u_1} {β : Type u_2} (S : Set α) (f : β ≃ α) : ↑f ⁻¹' S = ↑(Equiv.symm f) '' S

Alias for Equiv.image_eq_preimage

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theorem Equiv.subset_image {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : Set α) (t : Set β) : ↑(Equiv.symm e) '' t ⊆ s ↔ t ⊆ ↑e '' s

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theorem Equiv.subset_image' {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : Set α) (t : Set β) : s ⊆ ↑(Equiv.symm e) '' t ↔ ↑e '' s ⊆ t

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theorem Equiv.symm_image_image {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : Set α) : ↑(Equiv.symm e) '' (↑e '' s) = s

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theorem Equiv.eq_image_iff_symm_image_eq {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : Set α) (t : Set β) : t = ↑e '' s ↔ ↑(Equiv.symm e) '' t = s

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theorem Equiv.image_symm_image {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : Set β) : ↑e '' (↑(Equiv.symm e) '' s) = s

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theorem Equiv.image_preimage {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : Set β) : ↑e '' (↑e ⁻¹' s) = s

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theorem Equiv.preimage_image {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : Set α) : ↑e ⁻¹' (↑e '' s) = s

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theorem Equiv.image_compl {α : Type u_1} {β : Type u_2} (f : α ≃ β) (s : Set α) : ↑f '' sᶜ = (↑f '' s)ᶜ

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theorem Equiv.symm_preimage_preimage {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : Set β) : ↑(Equiv.symm e) ⁻¹' (↑e ⁻¹' s) = s

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theorem Equiv.preimage_symm_preimage {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : Set α) : ↑e ⁻¹' (↑(Equiv.symm e) ⁻¹' s) = s

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theorem Equiv.preimage_subset {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : Set β) (t : Set β) : ↑e ⁻¹' s ⊆ ↑e ⁻¹' t ↔ s ⊆ t

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theorem Equiv.image_subset {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : Set α) (t : Set α) : ↑e '' s ⊆ ↑e '' t ↔ s ⊆ t

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theorem Equiv.image_eq_iff_eq {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : Set α) (t : Set α) : ↑e '' s = ↑e '' t ↔ s = t

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theorem Equiv.preimage_eq_iff_eq_image {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : Set β) (t : Set α) : ↑e ⁻¹' s = t ↔ s = ↑e '' t

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theorem Equiv.eq_preimage_iff_image_eq {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : Set α) (t : Set β) : s = ↑e ⁻¹' t ↔ ↑e '' s = t

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theorem Equiv.prod_assoc_preimage {α : Type u_1} {β : Type u_2} {γ : Type u_3} {s : Set α} {t : Set β} {u : Set γ} : ↑(Equiv.prodAssoc α β γ) ⁻¹' s ×ˢ t ×ˢ u = (s ×ˢ t) ×ˢ u

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theorem Equiv.prod_assoc_symm_preimage {α : Type u_1} {β : Type u_2} {γ : Type u_3} {s : Set α} {t : Set β} {u : Set γ} : ↑(Equiv.symm (Equiv.prodAssoc α β γ)) ⁻¹' (s ×ˢ t) ×ˢ u = s ×ˢ t ×ˢ u

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theorem Equiv.prod_assoc_image {α : Type u_1} {β : Type u_2} {γ : Type u_3} {s : Set α} {t : Set β} {u : Set γ} : ↑(Equiv.prodAssoc α β γ) '' (s ×ˢ t) ×ˢ u = s ×ˢ t ×ˢ u

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theorem Equiv.prod_assoc_symm_image {α : Type u_1} {β : Type u_2} {γ : Type u_3} {s : Set α} {t : Set β} {u : Set γ} : ↑(Equiv.symm (Equiv.prodAssoc α β γ)) '' s ×ˢ t ×ˢ u = (s ×ˢ t) ×ˢ u

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def Equiv.setProdEquivSigma {α : Type u_1} {β : Type u_2} (s : Set (α × β)) : ↑s ≃ (x : α) × ↑{ y | (x, y) ∈ s }

A set s in α × β is equivalent to the sigma-type Σ x, {y | (x, y) ∈ s}.

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theorem Equiv.setCongr_apply {α : Type u_1} {s : Set α} {t : Set α} (h : s = t) (a : { a // (fun a => (fun x => x ∈ s) a) a }) : ↑(Equiv.setCongr h) a = { val := ↑a, property := (_ : ↑(Equiv.refl α) ↑a ∈ t) }

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def Equiv.setCongr {α : Type u_1} {s : Set α} {t : Set α} (h : s = t) : ↑s ≃ ↑t

The subtypes corresponding to equal sets are equivalent.

Equations

• Equiv.setCongr h = Equiv.subtypeEquivProp h

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theorem Equiv.image_symm_apply_coe {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : Set α) (y : ↑(↑e '' s)) : ↑(↑(Equiv.symm (Equiv.image e s)) y) = ↑(Equiv.symm e) ↑y

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theorem Equiv.image_apply_coe {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : Set α) (x : ↑s) : ↑(↑(Equiv.image e s) x) = ↑e ↑x

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def Equiv.image {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : Set α) : ↑s ≃ ↑(↑e '' s)

A set is equivalent to its image under an equivalence.

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def Equiv.Set.univ (α : Type u_1) : ↑Set.univ ≃ α

univ α is equivalent to α.

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def Equiv.Set.empty (α : Type u_1) : ↑∅ ≃ Empty

An empty set is equivalent to the Empty type.

Equations

• Equiv.Set.empty α = Equiv.equivEmpty ↑∅

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def Equiv.Set.pempty (α : Type u_1) : ↑∅ ≃ PEmpty

An empty set is equivalent to a PEmpty type.

Equations

• Equiv.Set.pempty α = Equiv.equivPEmpty ↑∅

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def Equiv.Set.union' {α : Type u_1} {s : Set α} {t : Set α} (p : α → Prop) [inst : DecidablePred p] (hs : (x : α) → x ∈ s → p x) (ht : ∀ (x : α), x ∈ t → ¬p x) : ↑(s ∪ t) ≃ ↑s ⊕ ↑t

If sets s and t are separated by a decidable predicate, then s ∪ t is equivalent to s ⊕ t.

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• One or more equations did not get rendered due to their size.

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def Equiv.Set.union {α : Type u_1} {s : Set α} {t : Set α} [inst : DecidablePred fun x => x ∈ s] (H : s ∩ t ⊆ ∅) : ↑(s ∪ t) ≃ ↑s ⊕ ↑t

If sets s and t are disjoint, then s ∪ t is equivalent to s ⊕ t.

Equations

• Equiv.Set.union H = Equiv.Set.union' (fun x => x ∈ s) (_ : ∀ (x : α), x ∈ s → x ∈ s) (_ : ∀ (x : α), x ∈ t → x ∈ s → x ∈ ∅)

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theorem Equiv.Set.union_apply_left {α : Type u_1} {s : Set α} {t : Set α} [inst : DecidablePred fun x => x ∈ s] (H : s ∩ t ⊆ ∅) {a : ↑(s ∪ t)} (ha : ↑a ∈ s) : ↑(Equiv.Set.union H) a = Sum.inl { val := ↑a, property := ha }

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theorem Equiv.Set.union_apply_right {α : Type u_1} {s : Set α} {t : Set α} [inst : DecidablePred fun x => x ∈ s] (H : s ∩ t ⊆ ∅) {a : ↑(s ∪ t)} (ha : ↑a ∈ t) : ↑(Equiv.Set.union H) a = Sum.inr { val := ↑a, property := ha }

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theorem Equiv.Set.union_symm_apply_left {α : Type u_1} {s : Set α} {t : Set α} [inst : DecidablePred fun x => x ∈ s] (H : s ∩ t ⊆ ∅) (a : ↑s) : ↑(Equiv.symm (Equiv.Set.union H)) (Sum.inl a) = { val := ↑a, property := (_ : ↑a ∈ s ∪ t) }

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theorem Equiv.Set.union_symm_apply_right {α : Type u_1} {s : Set α} {t : Set α} [inst : DecidablePred fun x => x ∈ s] (H : s ∩ t ⊆ ∅) (a : ↑t) : ↑(Equiv.symm (Equiv.Set.union H)) (Sum.inr a) = { val := ↑a, property := (_ : ↑a ∈ s ∪ t) }

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def Equiv.Set.singleton {α : Type u_1} (a : α) : ↑{a} ≃ PUnit

A singleton set is equivalent to a PUnit type.

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theorem Equiv.Set.ofEq_apply {α : Type u} {s : Set α} {t : Set α} (h : s = t) (a : { a // (fun a => (fun x => x ∈ s) a) a }) : ↑(Equiv.Set.ofEq h) a = { val := ↑a, property := (_ : ↑(Equiv.refl α) ↑a ∈ t) }

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theorem Equiv.Set.ofEq_symm_apply {α : Type u} {s : Set α} {t : Set α} (h : s = t) (b : { b // (fun b => (fun x => x ∈ t) b) b }) : ↑(Equiv.symm (Equiv.Set.ofEq h)) b = { val := ↑b, property := (_ : ↑(Equiv.symm (Equiv.refl α)) ↑b ∈ s) }

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def Equiv.Set.ofEq {α : Type u} {s : Set α} {t : Set α} (h : s = t) : ↑s ≃ ↑t

Equal sets are equivalent.

TODO: this is the same as Equiv.setCongr!

Equations

• Equiv.Set.ofEq h = Equiv.setCongr h

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def Equiv.Set.insert {α : Type u} {s : Set α} [inst : DecidablePred fun x => x ∈ s] {a : α} (H : ¬a ∈ s) : ↑(insert a s) ≃ ↑s ⊕ PUnit

If a ∉ s, then insert a s is equivalent to s ⊕ punit.

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theorem Equiv.Set.insert_symm_apply_inl {α : Type u} {s : Set α} [inst : DecidablePred fun x => x ∈ s] {a : α} (H : ¬a ∈ s) (b : ↑s) : ↑(Equiv.symm (Equiv.Set.insert H)) (Sum.inl b) = { val := ↑b, property := (_ : ↑b = a ∨ ↑b ∈ s) }

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theorem Equiv.Set.insert_symm_apply_inr {α : Type u} {s : Set α} [inst : DecidablePred fun x => x ∈ s] {a : α} (H : ¬a ∈ s) (b : PUnit) : ↑(Equiv.symm (Equiv.Set.insert H)) (Sum.inr b) = { val := a, property := (_ : a = a ∨ a ∈ s) }

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theorem Equiv.Set.insert_apply_left {α : Type u} {s : Set α} [inst : DecidablePred fun x => x ∈ s] {a : α} (H : ¬a ∈ s) : ↑(Equiv.Set.insert H) { val := a, property := (_ : a = a ∨ a ∈ s) } = Sum.inr PUnit.unit

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theorem Equiv.Set.insert_apply_right {α : Type u} {s : Set α} [inst : DecidablePred fun x => x ∈ s] {a : α} (H : ¬a ∈ s) (b : ↑s) : ↑(Equiv.Set.insert H) { val := ↑b, property := (_ : ↑b = a ∨ ↑b ∈ s) } = Sum.inl b

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def Equiv.Set.sumCompl {α : Type u_1} (s : Set α) [inst : DecidablePred fun x => x ∈ s] : ↑s ⊕ ↑(sᶜ) ≃ α

If s : Set α is a set with decidable membership, then s ⊕ sᶜ is equivalent to α.

Equations

• Equiv.Set.sumCompl s = Trans.trans (Trans.trans (Equiv.symm (Equiv.Set.union (_ : s ∩ sᶜ ⊆ ∅))) (Equiv.Set.ofEq (_ : s ∪ sᶜ = Set.univ))) (Equiv.Set.univ α)

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theorem Equiv.Set.sumCompl_apply_inl {α : Type u} (s : Set α) [inst : DecidablePred fun x => x ∈ s] (x : ↑s) : ↑(Equiv.Set.sumCompl s) (Sum.inl x) = ↑x

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theorem Equiv.Set.sumCompl_apply_inr {α : Type u} (s : Set α) [inst : DecidablePred fun x => x ∈ s] (x : ↑(sᶜ)) : ↑(Equiv.Set.sumCompl s) (Sum.inr x) = ↑x

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theorem Equiv.Set.sumCompl_symm_apply_of_mem {α : Type u} {s : Set α} [inst : DecidablePred fun x => x ∈ s] {x : α} (hx : x ∈ s) : ↑(Equiv.symm (Equiv.Set.sumCompl s)) x = Sum.inl { val := x, property := hx }

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theorem Equiv.Set.sumCompl_symm_apply_of_not_mem {α : Type u} {s : Set α} [inst : DecidablePred fun x => x ∈ s] {x : α} (hx : ¬x ∈ s) : ↑(Equiv.symm (Equiv.Set.sumCompl s)) x = Sum.inr { val := x, property := hx }

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theorem Equiv.Set.sumCompl_symm_apply {α : Type u_1} {s : Set α} [inst : DecidablePred fun x => x ∈ s] {x : ↑s} : ↑(Equiv.symm (Equiv.Set.sumCompl s)) ↑x = Sum.inl x

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theorem Equiv.Set.sumCompl_symm_apply_compl {α : Type u_1} {s : Set α} [inst : DecidablePred fun x => x ∈ s] {x : ↑(sᶜ)} : ↑(Equiv.symm (Equiv.Set.sumCompl s)) ↑x = Sum.inr x

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def Equiv.Set.sumDiffSubset {α : Type u_1} {s : Set α} {t : Set α} (h : s ⊆ t) [inst : DecidablePred fun x => x ∈ s] : ↑s ⊕ ↑(t \ s) ≃ ↑t

sumDiffSubset s t is the natural equivalence between s ⊕ (t \ s) and t, where s and t are two sets.

Equations

• Equiv.Set.sumDiffSubset h = Trans.trans (Equiv.symm (Equiv.Set.union (_ : s ∩ (t \ s) ⊆ ∅))) (Equiv.Set.ofEq (_ : s ∪ t \ s = t))

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theorem Equiv.Set.sumDiffSubset_apply_inl {α : Type u_1} {s : Set α} {t : Set α} (h : s ⊆ t) [inst : DecidablePred fun x => x ∈ s] (x : ↑s) : ↑(Equiv.Set.sumDiffSubset h) (Sum.inl x) = Set.inclusion h x

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theorem Equiv.Set.sumDiffSubset_apply_inr {α : Type u_1} {s : Set α} {t : Set α} (h : s ⊆ t) [inst : DecidablePred fun x => x ∈ s] (x : ↑(t \ s)) : ↑(Equiv.Set.sumDiffSubset h) (Sum.inr x) = Set.inclusion (_ : t \ s ⊆ t) x

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theorem Equiv.Set.sumDiffSubset_symm_apply_of_mem {α : Type u_1} {s : Set α} {t : Set α} (h : s ⊆ t) [inst : DecidablePred fun x => x ∈ s] {x : ↑t} (hx : ↑x ∈ s) : ↑(Equiv.symm (Equiv.Set.sumDiffSubset h)) x = Sum.inl { val := ↑x, property := hx }

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theorem Equiv.Set.sumDiffSubset_symm_apply_of_not_mem {α : Type u_1} {s : Set α} {t : Set α} (h : s ⊆ t) [inst : DecidablePred fun x => x ∈ s] {x : ↑t} (hx : ¬↑x ∈ s) : ↑(Equiv.symm (Equiv.Set.sumDiffSubset h)) x = Sum.inr { val := ↑x, property := (_ : ↑x ∈ t ∧ ¬↑x ∈ s) }

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def Equiv.Set.unionSumInter {α : Type u} (s : Set α) (t : Set α) [inst : DecidablePred fun x => x ∈ s] : ↑(s ∪ t) ⊕ ↑(s ∩ t) ≃ ↑s ⊕ ↑t

If s is a set with decidable membership, then the sum of s ∪ t and s ∩ t is equivalent to s ⊕ t.

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def Equiv.Set.compl {α : Type u} {β : Type v} {s : Set α} {t : Set β} [inst : DecidablePred fun x => x ∈ s] [inst : DecidablePred fun x => x ∈ t] (e₀ : ↑s ≃ ↑t) : { e // ∀ (x : ↑s), ↑e ↑x = ↑(↑e₀ x) } ≃ (↑(sᶜ) ≃ ↑(tᶜ))

Given an equivalence e₀ between sets s : Set α and t : Set β, the set of equivalences e : α ≃ β such that e ↑x = ↑(e₀ x) for each x : s is equivalent to the set of equivalences between sᶜ and tᶜ.

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• One or more equations did not get rendered due to their size.

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def Equiv.Set.prod {α : Type u_1} {β : Type u_2} (s : Set α) (t : Set β) : ↑(s ×ˢ t) ≃ ↑s × ↑t

The set product of two sets is equivalent to the type product of their coercions to types.

Equations

• Equiv.Set.prod s t = Equiv.subtypeProdEquivProd

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theorem Equiv.Set.univPi_symm_apply_coe {α : Type u_1} {β : α → Type u_2} (s : (a : α) → Set (β a)) (f : (a : α) → ↑(s a)) (a : α) : ↑(↑(Equiv.symm (Equiv.Set.univPi s)) f) a = ↑(f a)

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theorem Equiv.Set.univPi_apply_coe {α : Type u_1} {β : α → Type u_2} (s : (a : α) → Set (β a)) (f : ↑(Set.pi Set.univ s)) (a : α) : ↑(↑(Equiv.Set.univPi s) f a) = ↑f a

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def Equiv.Set.univPi {α : Type u_1} {β : α → Type u_2} (s : (a : α) → Set (β a)) : ↑(Set.pi Set.univ s) ≃ ((a : α) → ↑(s a))

The set Set.pi Set.univ s is equivalent to Π a, s a.

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• One or more equations did not get rendered due to their size.

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noncomputable def Equiv.Set.imageOfInjOn {α : Type u_1} {β : Type u_2} (f : α → β) (s : Set α) (H : Set.InjOn f s) : ↑s ≃ ↑(f '' s)

If a function f is injective on a set s, then s is equivalent to f '' s.

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• One or more equations did not get rendered due to their size.

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theorem Equiv.Set.image_apply {α : Type u_1} {β : Type u_2} (f : α → β) (s : Set α) (H : Function.Injective f) (p : ↑s) : ↑(Equiv.Set.image f s H) p = { val := f ↑p, property := (_ : f ↑p ∈ f '' s) }

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noncomputable def Equiv.Set.image {α : Type u_1} {β : Type u_2} (f : α → β) (s : Set α) (H : Function.Injective f) : ↑s ≃ ↑(f '' s)

If f is an injective function, then s is equivalent to f '' s.

Equations

• Equiv.Set.image f s H = Equiv.Set.imageOfInjOn f s (_ : Set.InjOn f s)

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theorem Equiv.Set.image_symm_apply {α : Type u_1} {β : Type u_2} (f : α → β) (s : Set α) (H : Function.Injective f) (x : α) (h : x ∈ s) : ↑(Equiv.symm (Equiv.Set.image f s H)) { val := f x, property := (_ : ∃ a, a ∈ s ∧ f a = f x) } = { val := x, property := h }

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theorem Equiv.Set.image_symm_preimage {α : Type u_1} {β : Type u_2} {f : α → β} (hf : Function.Injective f) (u : Set α) (s : Set α) : (fun x => ↑(↑(Equiv.symm (Equiv.Set.image f s hf)) x)) ⁻¹' u = Subtype.val ⁻¹' (f '' u)

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theorem Equiv.Set.congr_symm_apply {α : Type u_1} {β : Type u_2} (e : α ≃ β) (t : Set β) : ↑(Equiv.symm (Equiv.Set.congr e)) t = ↑(Equiv.symm e) '' t

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theorem Equiv.Set.congr_apply {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : Set α) : ↑(Equiv.Set.congr e) s = ↑e '' s

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def Equiv.Set.congr {α : Type u_1} {β : Type u_2} (e : α ≃ β) : Set α ≃ Set β

If α is equivalent to β, then Set α is equivalent to Set β.

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• One or more equations did not get rendered due to their size.

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def Equiv.Set.sep {α : Type u} (s : Set α) (t : α → Prop) : ↑{ x | x ∈ s ∧ t x } ≃ ↑{ x | t ↑x }

The set {x ∈ s | t x} is equivalent to the set of x : s such that t x.

Equations

• Equiv.Set.sep s t = Equiv.symm (Equiv.subtypeSubtypeEquivSubtypeInter s t)

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def Equiv.Set.powerset {α : Type u_1} (S : Set α) : ↑(𝒫 S) ≃ Set ↑S

The set 𝒫 S := {x | x ⊆ S} is equivalent to the type Set S.

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• One or more equations did not get rendered due to their size.

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theorem Equiv.Set.rangeSplittingImageEquiv_symm_apply_coe {α : Type u_1} {β : Type u_2} (f : α → β) (s : Set ↑(Set.range f)) (x : ↑s) : ↑(↑(Equiv.symm (Equiv.Set.rangeSplittingImageEquiv f s)) x) = Set.rangeSplitting f ↑x

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theorem Equiv.Set.rangeSplittingImageEquiv_apply_coe_coe {α : Type u_1} {β : Type u_2} (f : α → β) (s : Set ↑(Set.range f)) (x : ↑(Set.rangeSplitting f '' s)) : ↑↑(↑(Equiv.Set.rangeSplittingImageEquiv f s) x) = f ↑x

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noncomputable def Equiv.Set.rangeSplittingImageEquiv {α : Type u_1} {β : Type u_2} (f : α → β) (s : Set ↑(Set.range f)) : ↑(Set.rangeSplitting f '' s) ≃ ↑s

If s is a set in range f, then its image under rangeSplitting f is in bijection (via f) with s.

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• One or more equations did not get rendered due to their size.

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theorem Equiv.Set.rangeInl_symm_apply_coe (α : Type u_1) (β : Type u_2) (x : α) : ↑(↑(Equiv.symm (Equiv.Set.rangeInl α β)) x) = Sum.inl x

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def Equiv.Set.rangeInl (α : Type u_1) (β : Type u_2) : ↑(Set.range Sum.inl) ≃ α

Equivalence between the range of Sum.inl : α → α ⊕ β and α.

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• One or more equations did not get rendered due to their size.

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@[simp]

theorem Equiv.Set.rangeInl_apply_inl {α : Type u_1} (β : Type u_2) (x : α) : ↑(Equiv.Set.rangeInl α β) { val := Sum.inl x, property := (_ : Sum.inl x ∈ Set.range Sum.inl) } = x

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theorem Equiv.Set.rangeInr_symm_apply_coe (α : Type u_1) (β : Type u_2) (x : β) : ↑(↑(Equiv.symm (Equiv.Set.rangeInr α β)) x) = Sum.inr x

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def Equiv.Set.rangeInr (α : Type u_1) (β : Type u_2) : ↑(Set.range Sum.inr) ≃ β

Equivalence between the range of Sum.inr : β → α ⊕ β and β.

Equations

• One or more equations did not get rendered due to their size.

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@[simp]

theorem Equiv.Set.rangeInr_apply_inr (α : Type u_1) {β : Type u_2} (x : β) : ↑(Equiv.Set.rangeInr α β) { val := Sum.inr x, property := (_ : Sum.inr x ∈ Set.range Sum.inr) } = x

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@[simp]

theorem Equiv.ofLeftInverse_apply_coe {α : Sort u_1} {β : Type u_2} (f : α → β) (f_inv : Nonempty α → β → α) (hf : ∀ (h : Nonempty α), Function.LeftInverse (f_inv h) f) (a : α) : ↑(↑(Equiv.ofLeftInverse f f_inv hf) a) = f a

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theorem Equiv.ofLeftInverse_symm_apply {α : Sort u_1} {β : Type u_2} (f : α → β) (f_inv : Nonempty α → β → α) (hf : ∀ (h : Nonempty α), Function.LeftInverse (f_inv h) f) (b : ↑(Set.range f)) : ↑(Equiv.symm (Equiv.ofLeftInverse f f_inv hf)) b = f_inv (_ : Nonempty α) ↑b

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def Equiv.ofLeftInverse {α : Sort u_1} {β : Type u_2} (f : α → β) (f_inv : Nonempty α → β → α) (hf : ∀ (h : Nonempty α), Function.LeftInverse (f_inv h) f) : α ≃ ↑(Set.range f)

If f : α → β has a left-inverse when α is nonempty, then α is computably equivalent to the range of f.

While awkward, the Nonempty α hypothesis on f_inv and hf allows this to be used when α is empty too. This hypothesis is absent on analogous definitions on stronger Equivs like LinearEquiv.ofLeftInverse and RingEquiv.ofLeftInverse as their typeclass assumptions are already sufficient to ensure non-emptiness.

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• One or more equations did not get rendered due to their size.

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@[inline]

abbrev Equiv.ofLeftInverse' {α : Sort u_1} {β : Type u_2} (f : α → β) (f_inv : β → α) (hf : Function.LeftInverse f_inv f) : α ≃ ↑(Set.range f)

If f : α → β has a left-inverse, then α is computably equivalent to the range of f.

Note that if α is empty, no such f_inv exists and so this definition can't be used, unlike the stronger but less convenient ofLeftInverse.

Equations

• Equiv.ofLeftInverse' f f_inv hf = Equiv.ofLeftInverse f (fun x => f_inv) (_ : Nonempty α → Function.LeftInverse f_inv f)

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theorem Equiv.ofInjective_apply {α : Sort u_1} {β : Type u_2} (f : α → β) (hf : Function.Injective f) (a : α) : ↑(Equiv.ofInjective f hf) a = { val := f a, property := (_ : ∃ y, f y = f a) }

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noncomputable def Equiv.ofInjective {α : Sort u_1} {β : Type u_2} (f : α → β) (hf : Function.Injective f) : α ≃ ↑(Set.range f)

If f : α → β is an injective function, then domain α is equivalent to the range of f.

Equations

• Equiv.ofInjective f hf = Equiv.ofLeftInverse f (fun x => Function.invFun f) (_ : ∀ (x : Nonempty α), Function.LeftInverse (Function.invFun f) f)

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theorem Equiv.apply_ofInjective_symm {α : Sort u_1} {β : Type u_2} {f : α → β} (hf : Function.Injective f) (b : ↑(Set.range f)) : f (↑(Equiv.symm (Equiv.ofInjective f hf)) b) = ↑b

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@[simp]

theorem Equiv.ofInjective_symm_apply {α : Sort u_1} {β : Type u_2} {f : α → β} (hf : Function.Injective f) (a : α) : ↑(Equiv.symm (Equiv.ofInjective f hf)) { val := f a, property := (_ : ∃ y, f y = f a) } = a

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theorem Equiv.coe_ofInjective_symm {α : Type u_1} {β : Type u_2} {f : α → β} (hf : Function.Injective f) : ↑(Equiv.symm (Equiv.ofInjective f hf)) = Set.rangeSplitting f

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theorem Equiv.self_comp_ofInjective_symm {α : Sort u_1} {β : Type u_2} {f : α → β} (hf : Function.Injective f) : f ∘ ↑(Equiv.symm (Equiv.ofInjective f hf)) = Subtype.val

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theorem Equiv.ofLeftInverse_eq_ofInjective {α : Type u_1} {β : Type u_2} (f : α → β) (f_inv : Nonempty α → β → α) (hf : ∀ (h : Nonempty α), Function.LeftInverse (f_inv h) f) : Equiv.ofLeftInverse f f_inv hf = Equiv.ofInjective f (_ : Function.Injective f)

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theorem Equiv.ofLeftInverse'_eq_ofInjective {α : Type u_1} {β : Type u_2} (f : α → β) (f_inv : β → α) (hf : Function.LeftInverse f_inv f) : Equiv.ofLeftInverse' f f_inv hf = Equiv.ofInjective f (_ : Function.Injective f)

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theorem Equiv.set_forall_iff {α : Type u_1} {β : Type u_2} (e : α ≃ β) {p : Set α → Prop} : ((a : Set α) → p a) ↔ (a : Set β) → p (↑e ⁻¹' a)

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theorem Equiv.preimage_piEquivPiSubtypeProd_symm_pi {α : Type u_1} {β : α → Type u_2} (p : α → Prop) [inst : DecidablePred p] (s : (i : α) → Set (β i)) : ↑(Equiv.symm (Equiv.piEquivPiSubtypeProd p β)) ⁻¹' Set.pi Set.univ s = (Set.pi Set.univ fun i => s ↑i) ×ˢ Set.pi Set.univ fun i => s ↑i

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@[simp]

theorem Equiv.sigmaPreimageEquiv_symm_apply_snd_coe {α : Type u_1} {β : Type u_2} (f : α → β) (x : α) : ↑(↑(Equiv.symm (Equiv.sigmaPreimageEquiv f)) x).snd = x

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theorem Equiv.sigmaPreimageEquiv_apply {α : Type u_1} {β : Type u_2} (f : α → β) (x : (y : β) × { x // f x = y }) : ↑(Equiv.sigmaPreimageEquiv f) x = ↑x.snd

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theorem Equiv.sigmaPreimageEquiv_symm_apply_fst {α : Type u_1} {β : Type u_2} (f : α → β) (x : α) : (↑(Equiv.symm (Equiv.sigmaPreimageEquiv f)) x).fst = f x

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def Equiv.sigmaPreimageEquiv {α : Type u_1} {β : Type u_2} (f : α → β) : (b : β) × ↑(f ⁻¹' {b}) ≃ α

sigmaPreimageEquiv f for f : α → β is the natural equivalence between the type of all preimages of points under f and the total space α.

Equations

• Equiv.sigmaPreimageEquiv f = Equiv.sigmaFiberEquiv f

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theorem Equiv.ofPreimageEquiv_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → γ} {g : β → γ} (e : (c : γ) → ↑(f ⁻¹' {c}) ≃ ↑(g ⁻¹' {c})) : ∀ (a : α), ↑(Equiv.ofPreimageEquiv e) a = ↑(↑(e (f a)) (↑(Equiv.symm (Equiv.sigmaFiberEquiv fun a => f a)) a).snd)

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theorem Equiv.ofPreimageEquiv_symm_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → γ} {g : β → γ} (e : (c : γ) → ↑(f ⁻¹' {c}) ≃ ↑(g ⁻¹' {c})) : ∀ (a : β), ↑(Equiv.symm (Equiv.ofPreimageEquiv e)) a = ↑(↑(Equiv.symm (Equiv.sigmaCongrRight e)) (↑(Equiv.symm (Equiv.sigmaFiberEquiv fun b => g b)) a)).snd

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def Equiv.ofPreimageEquiv {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → γ} {g : β → γ} (e : (c : γ) → ↑(f ⁻¹' {c}) ≃ ↑(g ⁻¹' {c})) : α ≃ β

A family of equivalences between preimages of points gives an equivalence between domains.

Equations

• Equiv.ofPreimageEquiv e = Equiv.ofFiberEquiv e

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theorem Equiv.ofPreimageEquiv_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → γ} {g : β → γ} (e : (c : γ) → ↑(f ⁻¹' {c}) ≃ ↑(g ⁻¹' {c})) (a : α) : g (↑(Equiv.ofPreimageEquiv e) a) = f a

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noncomputable def Set.BijOn.equiv {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} (f : α → β) (h : Set.BijOn f s t) : ↑s ≃ ↑t

If a function is a bijection between two sets s and t, then it induces an equivalence between the types ↥s and ↥t.

Equations

• Set.BijOn.equiv f h = Equiv.ofBijective (Set.MapsTo.restrict f s t (_ : Set.MapsTo f s t)) (_ : Function.Bijective (Set.MapsTo.restrict f s t (_ : Set.MapsTo f s t)))

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theorem dite_comp_equiv_update {α : Type u_1} {β : Sort u_2} {γ : Sort u_3} {s : Set α} (e : β ≃ ↑s) (v : β → γ) (w : α → γ) (j : β) (x : γ) [inst : DecidableEq β] [inst : DecidableEq α] [inst : (j : α) → Decidable (j ∈ s)] : (fun i => if h : i ∈ s then Function.update v j x (↑(Equiv.symm e) { val := i, property := h }) else w i) = Function.update (fun i => if h : i ∈ s then v (↑(Equiv.symm e) { val := i, property := h }) else w i) (↑(↑e j)) x

The composition of an updated function with an equiv on a subset can be expressed as an updated function.