Theorem homafval 17660

Description: Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
homarcl.h	⊢ 𝐻 = (Homa‘𝐶)
homafval.b	⊢ 𝐵 = (Base‘𝐶)
homafval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
homafval.j	⊢ 𝐽 = (Hom ‘𝐶)

Assertion

Ref	Expression
homafval	⊢ (𝜑 → 𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥))))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥

Allowed substitution hints:   𝐻(𝑥)   𝐽(𝑥)

Proof of Theorem homafval

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		homarcl.h	. 2 ⊢ 𝐻 = (Homa‘𝐶)
2		homafval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		fveq2 6756	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4		homafval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐶)
5	3, 4	eqtr4di 2797	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6	5	sqxpeqd 5612	. . . . 5 ⊢ (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵))
7		fveq2 6756	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
8		homafval.j	. . . . . . . 8 ⊢ 𝐽 = (Hom ‘𝐶)
9	7, 8	eqtr4di 2797	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐽)
10	9	fveq1d 6758	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((Hom ‘𝑐)‘𝑥) = (𝐽‘𝑥))
11	10	xpeq2d 5610	. . . . 5 ⊢ (𝑐 = 𝐶 → ({𝑥} × ((Hom ‘𝑐)‘𝑥)) = ({𝑥} × (𝐽‘𝑥)))
12	6, 11	mpteq12dv 5161	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥))))
13		df-homa 17657	. . . 4 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
14	4	fvexi 6770	. . . . . 6 ⊢ 𝐵 ∈ V
15	14, 14	xpex 7581	. . . . 5 ⊢ (𝐵 × 𝐵) ∈ V
16	15	mptex 7081	. . . 4 ⊢ (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥))) ∈ V
17	12, 13, 16	fvmpt 6857	. . 3 ⊢ (𝐶 ∈ Cat → (Homa‘𝐶) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥))))
18	2, 17	syl 17	. 2 ⊢ (𝜑 → (Homa‘𝐶) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥))))
19	1, 18	eqtrid 2790	1 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥))))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  {csn 4558   ↦ cmpt 5153   × cxp 5578  ‘cfv 6418  Basecbs 16840  Hom chom 16899  Catccat 17290  Hom_achoma 17654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-homa 17657

This theorem is referenced by:  homaf  17661  homaval  17662