Documentation

Mathlib.Data.Set.Opposite

The opposite of a set #

The opposite of a set s is simply the set obtained by taking the opposite of each member of s.

def Set.op {α : Type u_1} (s : Set α) :

The opposite of a set s is the set obtained by taking the opposite of each member of s.

Equations
def Set.unop {α : Type u_1} (s : Set αᵒᵖ) :
Set α

The unop of a set s is the set obtained by taking the unop of each member of s.

Equations
@[simp]
theorem Set.mem_op {α : Type u_1} {s : Set α} {a : αᵒᵖ} :
@[simp]
theorem Set.op_mem_op {α : Type u_1} {s : Set α} {a : α} :
@[simp]
theorem Set.mem_unop {α : Type u_1} {s : Set αᵒᵖ} {a : α} :
@[simp]
theorem Set.unop_mem_unop {α : Type u_1} {s : Set αᵒᵖ} {a : αᵒᵖ} :
@[simp]
theorem Set.op_unop {α : Type u_1} (s : Set α) :
@[simp]
theorem Set.unop_op {α : Type u_1} (s : Set αᵒᵖ) :
@[simp]
theorem Set.opEquiv_self_symm_apply_coe {α : Type u_1} (s : Set α) (x : s) :
@[simp]
theorem Set.opEquiv_self_apply_coe {α : Type u_1} (s : Set α) (x : ↑(Set.op s)) :
↑(↑(Set.opEquiv_self s) x) = Opposite.unop x
def Set.opEquiv_self {α : Type u_1} (s : Set α) :
↑(Set.op s) s

The members of the opposite of a set are in bijection with the members of the set itself.

Equations
  • One or more equations did not get rendered due to their size.
@[simp]
theorem Set.opEquiv_apply {α : Type u_1} (s : Set α) :
Set.opEquiv s = Set.op s
@[simp]
theorem Set.opEquiv_symm_apply {α : Type u_1} (s : Set αᵒᵖ) :
↑(Equiv.symm Set.opEquiv) s = Set.unop s
def Set.opEquiv {α : Type u_1} :

Taking opposites as an equivalence of powersets.

Equations
@[simp]
theorem Set.singleton_op {α : Type u_1} (x : α) :
@[simp]
theorem Set.singleton_unop {α : Type u_1} (x : αᵒᵖ) :
@[simp]
theorem Set.singleton_op_unop {α : Type u_1} (x : α) :
@[simp]
theorem Set.singleton_unop_op {α : Type u_1} (x : αᵒᵖ) :