Theorem homfval

Description: Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017)

	Ref	Expression
Hypotheses	homffval.f	⊢ 𝐹 = ( Homf ‘ 𝐶 )
	homffval.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	homffval.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	homfval.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	homfval.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	homfval	⊢ ( 𝜑 → ( 𝑋 𝐹 𝑌 ) = ( 𝑋 𝐻 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		homffval.f	⊢ 𝐹 = ( Homf ‘ 𝐶 )
2		homffval.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		homffval.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
4		homfval.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5		homfval.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6	1 2 3	homffval	⊢ 𝐹 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 𝐻 𝑦 ) )
7	6	a1i	⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 𝐻 𝑦 ) ) )
8		oveq12	⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( 𝑥 𝐻 𝑦 ) = ( 𝑋 𝐻 𝑌 ) )
9	8	adantl	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑥 𝐻 𝑦 ) = ( 𝑋 𝐻 𝑌 ) )
10		ovexd	⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) ∈ V )
11	7 9 4 5 10	ovmpod	⊢ ( 𝜑 → ( 𝑋 𝐹 𝑌 ) = ( 𝑋 𝐻 𝑌 ) )