Theorem homafval 17996

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 6840	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4		homafval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐶)
5	3, 4	eqtr4di 2789	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6	5	sqxpeqd 5663	. . . . 5 ⊢ (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵))
7		fveq2 6840	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
8		homafval.j	. . . . . . . 8 ⊢ 𝐽 = (Hom ‘𝐶)
9	7, 8	eqtr4di 2789	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐽)
10	9	fveq1d 6842	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((Hom ‘𝑐)‘𝑥) = (𝐽‘𝑥))
11	10	xpeq2d 5661	. . . . 5 ⊢ (𝑐 = 𝐶 → ({𝑥} × ((Hom ‘𝑐)‘𝑥)) = ({𝑥} × (𝐽‘𝑥)))
12	6, 11	mpteq12dv 5172	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥))))
13		df-homa 17993	. . . 4 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
14	4	fvexi 6854	. . . . . 6 ⊢ 𝐵 ∈ V
15	14, 14	xpex 7707	. . . . 5 ⊢ (𝐵 × 𝐵) ∈ V
16	15	mptex 7178	. . . 4 ⊢ (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥))) ∈ V
17	12, 13, 16	fvmpt 6947	. . 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 2783	1 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥))))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {csn 4567   ↦ cmpt 5166   × cxp 5629  ‘cfv 6498  Basecbs 17179  Hom chom 17231  Catccat 17630  Hom_achoma 17990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-homa 17993

This theorem is referenced by:  homaf  17997  homaval  17998