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

      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) => Opposite.unop x, , invFun := fun (x : s) => Opposite.op x, , left_inv := , right_inv := }
      Instances For
        @[simp]
        theorem Set.opEquiv_apply {α : Type u_1} (s : Set α) :
        Set.opEquiv s = s.op
        @[simp]
        theorem Set.opEquiv_symm_apply {α : Type u_1} (s : Set αᵒᵖ) :
        Set.opEquiv.symm s = s.unop
        def Set.opEquiv {α : Type u_1} :

        Taking opposites as an equivalence of powersets.

        Equations
        • Set.opEquiv = { toFun := Set.op, invFun := Set.unop, left_inv := , right_inv := }
        Instances For
          @[simp]
          theorem Set.singleton_op {α : Type u_1} (x : α) :
          {x}.op = {Opposite.op x}
          @[simp]
          theorem Set.singleton_unop {α : Type u_1} (x : αᵒᵖ) :
          {x}.unop = {Opposite.unop x}
          @[simp]
          theorem Set.singleton_op_unop {α : Type u_1} (x : α) :
          {Opposite.op x}.unop = {x}
          @[simp]
          theorem Set.singleton_unop_op {α : Type u_1} (x : αᵒᵖ) :
          {Opposite.unop x}.op = {x}