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[Merged by Bors] - feat: add Equiv.prodSubtypeFstEquivSubtypeProd #12802

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[Merged by Bors] - feat: add Equiv.prodSubtypeFstEquivSubtypeProd #12802

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xroblot May 10, 2024
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Update Mathlib/Logic/Equiv/Basic.lean
xroblot May 10, 2024
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8 changes: 8 additions & 0 deletions Mathlib/Logic/Equiv/Basic.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -1380,6 +1380,14 @@ def subtypeProdEquivProd {p : α → Prop} {q : β → Prop} :
right_inv := fun ⟨⟨_, _⟩, ⟨_, _⟩⟩ => rfl
#align equiv.subtype_prod_equiv_prod Equiv.subtypeProdEquivProd

/-- A subtype of a `Prod` that depends only on the first component is equivalent to the
corresponding subtype of the first type times the second type. -/
def prodSubtypeFstEquivSubtypeProd {p : α → Prop} : {s : α × β // p s.1} ≃ {a // p a} × β where
toFun x := ⟨⟨x.1.1, x.2⟩, x.1.2⟩
invFun x := ⟨⟨x.1.1, x.2⟩, x.1.2⟩
left_inv _ := rfl
right_inv _ := rfl

/-- A subtype of a `Prod` is equivalent to a sigma type whose fibers are subtypes. -/
def subtypeProdEquivSigmaSubtype (p : α → β → Prop) :
{ x : α × β // p x.1 x.2 } ≃ Σa, { b : β // p a b } where
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