Theorem homafval 17744

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 6774	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4		homafval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐶)
5	3, 4	eqtr4di 2796	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6	5	sqxpeqd 5621	. . . . 5 ⊢ (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵))
7		fveq2 6774	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
8		homafval.j	. . . . . . . 8 ⊢ 𝐽 = (Hom ‘𝐶)
9	7, 8	eqtr4di 2796	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐽)
10	9	fveq1d 6776	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((Hom ‘𝑐)‘𝑥) = (𝐽‘𝑥))
11	10	xpeq2d 5619	. . . . 5 ⊢ (𝑐 = 𝐶 → ({𝑥} × ((Hom ‘𝑐)‘𝑥)) = ({𝑥} × (𝐽‘𝑥)))
12	6, 11	mpteq12dv 5165	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥))))
13		df-homa 17741	. . . 4 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
14	4	fvexi 6788	. . . . . 6 ⊢ 𝐵 ∈ V
15	14, 14	xpex 7603	. . . . 5 ⊢ (𝐵 × 𝐵) ∈ V
16	15	mptex 7099	. . . 4 ⊢ (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥))) ∈ V
17	12, 13, 16	fvmpt 6875	. . 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 2106  {csn 4561   ↦ cmpt 5157   × cxp 5587  ‘cfv 6433  Basecbs 16912  Hom chom 16973  Catccat 17373  Hom_achoma 17738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-homa 17741

This theorem is referenced by:  homaf  17745  homaval  17746