Theorem catchom

Description: Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017)

	Ref	Expression
Hypotheses	catcbas.c	⊢ 𝐶 = ( CatCat ‘ 𝑈 )
	catcbas.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	catcbas.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
	catchomfval.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	catchom.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	catchom.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	catchom	⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝑋 Func 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		catcbas.c	⊢ 𝐶 = ( CatCat ‘ 𝑈 )
2		catcbas.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		catcbas.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
4		catchomfval.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
5		catchom.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		catchom.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7	1 2 3 4	catchomfval	⊢ ( 𝜑 → 𝐻 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 Func 𝑦 ) ) )
8		oveq12	⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( 𝑥 Func 𝑦 ) = ( 𝑋 Func 𝑌 ) )
9	8	adantl	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑥 Func 𝑦 ) = ( 𝑋 Func 𝑌 ) )
10		ovexd	⊢ ( 𝜑 → ( 𝑋 Func 𝑌 ) ∈ V )
11	7 9 5 6 10	ovmpod	⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝑋 Func 𝑌 ) )