Theorem homaf 17536

Description: Functionality 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)

Assertion

Ref	Expression
homaf	⊢ (𝜑 → 𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))

Proof of Theorem homaf

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		homarcl.h	. . 3 ⊢ 𝐻 = (Homa‘𝐶)
2		homafval.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
3		homafval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		eqid 2737	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
5	1, 2, 3, 4	homafval 17535	. 2 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥))))
6		snssi 4721	. . . . 5 ⊢ (𝑥 ∈ (𝐵 × 𝐵) → {𝑥} ⊆ (𝐵 × 𝐵))
7	6	adantl 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 × 𝐵)) → {𝑥} ⊆ (𝐵 × 𝐵))
8		ssv 3925	. . . 4 ⊢ ((Hom ‘𝐶)‘𝑥) ⊆ V
9		xpss12 5566	. . . 4 ⊢ (({𝑥} ⊆ (𝐵 × 𝐵) ∧ ((Hom ‘𝐶)‘𝑥) ⊆ V) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
10	7, 8, 9	sylancl 589	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 × 𝐵)) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
11		snex 5324	. . . . 5 ⊢ {𝑥} ∈ V
12		fvex 6730	. . . . 5 ⊢ ((Hom ‘𝐶)‘𝑥) ∈ V
13	11, 12	xpex 7538	. . . 4 ⊢ ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ V
14	13	elpw 4517	. . 3 ⊢ (({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ 𝒫 ((𝐵 × 𝐵) × V) ↔ ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
15	10, 14	sylibr 237	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 × 𝐵)) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ 𝒫 ((𝐵 × 𝐵) × V))
16	5, 15	fmpt3d 6933	1 ⊢ (𝜑 → 𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2110  Vcvv 3408   ⊆ wss 3866  𝒫 cpw 4513  {csn 4541   × cxp 5549  ⟶wf 6376  ‘cfv 6380  Basecbs 16760  Hom chom 16813  Catccat 17167  Hom_achoma 17529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-homa 17532

This theorem is referenced by:  homarcl2  17541  homarel  17542  arwhoma  17551