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) : Set αᵒᵖ

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

Equations

• Set.op s = Opposite.unop ⁻¹' s

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

• Set.unop s = Opposite.op ⁻¹' s

@[simp]

theorem Set.mem_op {α : Type u_1} {s : Set α} {a : αᵒᵖ} : a ∈ Set.op s ↔ a.unop ∈ s

@[simp]

theorem Set.op_mem_op {α : Type u_1} {s : Set α} {a : α} : Opposite.op a ∈ Set.op s ↔ a ∈ s

@[simp]

theorem Set.mem_unop {α : Type u_1} {s : Set αᵒᵖ} {a : α} : a ∈ Set.unop s ↔ Opposite.op a ∈ s

@[simp]

theorem Set.unop_mem_unop {α : Type u_1} {s : Set αᵒᵖ} {a : αᵒᵖ} : a.unop ∈ Set.unop s ↔ a ∈ s

@[simp]

theorem Set.op_unop {α : Type u_1} (s : Set α) : Set.unop (Set.op s) = s

@[simp]

theorem Set.unop_op {α : Type u_1} (s : Set αᵒᵖ) : Set.op (Set.unop s) = s

@[simp]

theorem Set.opEquiv_self_symm_apply_coe {α : Type u_1} (s : Set α) (x : ↑s) : ↑((Set.opEquiv_self s).symm x) = Opposite.op ↑x

@[simp]

theorem Set.opEquiv_self_apply_coe {α : Type u_1} (s : Set α) (x : ↑(Set.op s)) : ↑((Set.opEquiv_self s) x) = (↑x).unop

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) : Set.opEquiv.symm s = Set.unop s

def Set.opEquiv {α : Type u_1} : Set α ≃ Set αᵒᵖ

Taking opposites as an equivalence of powersets.

Equations

• Set.opEquiv = { toFun := Set.op, invFun := Set.unop, left_inv := (_ : ∀ (s : Set α), Set.unop (Set.op s) = s), right_inv := (_ : ∀ (s : Set αᵒᵖ), Set.op (Set.unop s) = s) }

@[simp]

theorem Set.singleton_op {α : Type u_1} (x : α) : Set.op {x} = {Opposite.op x}

@[simp]

theorem Set.singleton_unop {α : Type u_1} (x : αᵒᵖ) : Set.unop {x} = {x.unop}

@[simp]

theorem Set.singleton_op_unop {α : Type u_1} (x : α) : Set.unop {Opposite.op x} = {x}

@[simp]

theorem Set.singleton_unop_op {α : Type u_1} (x : αᵒᵖ) : Set.op {x.unop} = {x}