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Metamath Proof Explorer

Theorem elcatchom

Description: A morphism of the category of categories (in a universe) is a functor. See df-catc for the definition of the category Cat, which consists of all categories in the universe u (i.e., " u -small categories", see Definition 3.44. of Adamek p. 39), with functors as the morphisms ( catchom ). (Contributed by Zhi Wang, 14-Nov-2025)

Ref Expression
Hypotheses catcrcl.c ⊢ 𝐶 = ( CatCat ‘ 𝑈 )
catcrcl.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
catcrcl.f ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
Assertion elcatchom ( 𝜑 → 𝐹 ∈ ( 𝑋 Func 𝑌 ) )

Proof

Step Hyp Ref Expression
1 catcrcl.c ⊢ 𝐶 = ( CatCat ‘ 𝑈 )
2 catcrcl.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
3 catcrcl.f ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
4 eqid ⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
5 1 2 3 catcrcl ⊢ ( 𝜑 → 𝑈 ∈ V )
6 1 2 3 4 catcrcl2 ⊢ ( 𝜑 → ( 𝑋 ∈ ( Base ‘ 𝐶 ) ∧ 𝑌 ∈ ( Base ‘ 𝐶 ) ) )
7 6 simpld ⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝐶 ) )
8 6 simprd ⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐶 ) )
9 1 4 5 2 7 8 catchom ⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝑋 Func 𝑌 ) )
10 3 9 eleqtrd ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 Func 𝑌 ) )