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.

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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.

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    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.

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    Instances For
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      @[simp]
      theorem Set.mem_op {α : Type u_1} {s : Set α} {a : αᵒᵖ} :
      a ∈ s.op ↔ Opposite.unop a ∈ s
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      @[simp]
      theorem Set.op_mem_op {α : Type u_1} {s : Set α} {a : α} :
      Opposite.op a ∈ s.op ↔ a ∈ s
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      @[simp]
      theorem Set.mem_unop {α : Type u_1} {s : Set αᵒᵖ} {a : α} :
      a ∈ s.unop ↔ Opposite.op a ∈ s
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      @[simp]
      theorem Set.unop_mem_unop {α : Type u_1} {s : Set αᵒᵖ} {a : αᵒᵖ} :
      Opposite.unop a ∈ s.unop ↔ a ∈ s
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      @[simp]
      theorem Set.op_unop {α : Type u_1} (s : Set α) :
      s.op.unop = s
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      @[simp]
      theorem Set.unop_op {α : Type u_1} (s : Set αᵒᵖ) :
      s.unop.op = s
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      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
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        @[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
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        @[simp]
        theorem Set.opEquiv_self_apply_coe {α : Type u_1} (s : Set α) (x : ↑s.op) :
        ↑(s.opEquiv_self x) = Opposite.unop ↑x
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        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 := ⋯, right_inv := ⋯ }
        Instances For
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          @[simp]
          theorem Set.opEquiv_apply {α : Type u_1} (s : Set α) :
          Set.opEquiv s = s.op
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          @[simp]
          theorem Set.opEquiv_symm_apply {α : Type u_1} (s : Set αᵒᵖ) :
          Set.opEquiv.symm s = s.unop
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          @[simp]
          theorem Set.singleton_op {α : Type u_1} (x : α) :
          {x}.op = {Opposite.op x}
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          @[simp]
          theorem Set.singleton_unop {α : Type u_1} (x : αᵒᵖ) :
          {x}.unop = {Opposite.unop x}
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          @[simp]
          theorem Set.singleton_op_unop {α : Type u_1} (x : α) :
          {Opposite.op x}.unop = {x}
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          @[simp]
          theorem Set.singleton_unop_op {α : Type u_1} (x : αᵒᵖ) :
          {Opposite.unop x}.op = {x}