Theorems · Theorem · category theory

CategoryTheory.Limits.Concrete.prodEquiv_apply_fst

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {FC : C → C → Type u_1} {CC : C → Type w}
  [inst_1 : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)] [inst_2 : CategoryTheory.ConcreteCategory C FC] (X₁ X₂ : C)
  [inst_3 : CategoryTheory.Limits.HasBinaryProduct X₁ X₂]
  [inst_4 : CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.pair X₁ X₂) (CategoryTheory.forget C)]
  (x : CategoryTheory.ToType (X₁ ⨯ X₂)),
  ((CategoryTheory.Limits.Concrete.prodEquiv X₁ X₂) x).1 =
    (CategoryTheory.ConcreteCategory.hom CategoryTheory.Limits.prod.fst) x

Defined in

Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory

Cited by

1 results in Mathlib

Foundations

Depth 40 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryFunLikeCategoryTheory.ConcreteCategoryCategoryTheory.Limits.HasBinaryProductCategoryTheory.Limits.PreservesLimit