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
.
@[simp]
theorem
Set.opEquiv_self_symm_apply_coe
{α : Type u_1}
(s : Set α)
(x : ↑s)
:
↑(s.opEquiv_self.symm x) = { unop := ↑x }
@[simp]
theorem
Set.opEquiv_self_apply_coe
{α : Type u_1}
(s : Set α)
(x : ↑s.op)
:
↑(s.opEquiv_self x) = (↑x).unop
The members of the opposite of a set are in bijection with the members of the set itself.
Equations
- s.opEquiv_self = { toFun := fun (x : ↑s.op) => ⟨(↑x).unop, ⋯⟩, invFun := fun (x : ↑s) => ⟨{ unop := ↑x }, ⋯⟩, left_inv := ⋯, right_inv := ⋯ }