Theorem homffval

Description: Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017) (Proof shortened by AV, 1-Mar-2024)

	Ref	Expression
Hypotheses	homffval.f	⊢ 𝐹 = ( Homf ‘ 𝐶 )
	homffval.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	homffval.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
Assertion	homffval	⊢ 𝐹 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 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		fveq2	⊢ ( 𝑐 = 𝐶 → ( Base ‘ 𝑐 ) = ( Base ‘ 𝐶 ) )
5	4 2	eqtr4di	⊢ ( 𝑐 = 𝐶 → ( Base ‘ 𝑐 ) = 𝐵 )
6		fveq2	⊢ ( 𝑐 = 𝐶 → ( Hom ‘ 𝑐 ) = ( Hom ‘ 𝐶 ) )
7	6 3	eqtr4di	⊢ ( 𝑐 = 𝐶 → ( Hom ‘ 𝑐 ) = 𝐻 )
8	7	oveqd	⊢ ( 𝑐 = 𝐶 → ( 𝑥 ( Hom ‘ 𝑐 ) 𝑦 ) = ( 𝑥 𝐻 𝑦 ) )
9	5 5 8	mpoeq123dv	⊢ ( 𝑐 = 𝐶 → ( 𝑥 ∈ ( Base ‘ 𝑐 ) , 𝑦 ∈ ( Base ‘ 𝑐 ) ↦ ( 𝑥 ( Hom ‘ 𝑐 ) 𝑦 ) ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 𝐻 𝑦 ) ) )
10		df-homf	⊢ Homf = ( 𝑐 ∈ V ↦ ( 𝑥 ∈ ( Base ‘ 𝑐 ) , 𝑦 ∈ ( Base ‘ 𝑐 ) ↦ ( 𝑥 ( Hom ‘ 𝑐 ) 𝑦 ) ) )
11	2	fvexi	⊢ 𝐵 ∈ V
12	11 11	mpoex	⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 𝐻 𝑦 ) ) ∈ V
13	9 10 12	fvmpt	⊢ ( 𝐶 ∈ V → ( Homf ‘ 𝐶 ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 𝐻 𝑦 ) ) )
14		fvprc	⊢ ( ¬ 𝐶 ∈ V → ( Homf ‘ 𝐶 ) = ∅ )
15		fvprc	⊢ ( ¬ 𝐶 ∈ V → ( Base ‘ 𝐶 ) = ∅ )
16	2 15	syl5eq	⊢ ( ¬ 𝐶 ∈ V → 𝐵 = ∅ )
17	16	olcd	⊢ ( ¬ 𝐶 ∈ V → ( 𝐵 = ∅ ∨ 𝐵 = ∅ ) )
18		0mpo0	⊢ ( ( 𝐵 = ∅ ∨ 𝐵 = ∅ ) → ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 𝐻 𝑦 ) ) = ∅ )
19	17 18	syl	⊢ ( ¬ 𝐶 ∈ V → ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 𝐻 𝑦 ) ) = ∅ )
20	14 19	eqtr4d	⊢ ( ¬ 𝐶 ∈ V → ( Homf ‘ 𝐶 ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 𝐻 𝑦 ) ) )
21	13 20	pm2.61i	⊢ ( Homf ‘ 𝐶 ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 𝐻 𝑦 ) )
22	1 21	eqtri	⊢ 𝐹 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 𝐻 𝑦 ) )