Theorem homaf 17119

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 2795	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
5	1, 2, 3, 4	homafval 17118	. 2 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥))))
6		snssi 4648	. . . . 5 ⊢ (𝑥 ∈ (𝐵 × 𝐵) → {𝑥} ⊆ (𝐵 × 𝐵))
7	6	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 × 𝐵)) → {𝑥} ⊆ (𝐵 × 𝐵))
8		ssv 3912	. . . 4 ⊢ ((Hom ‘𝐶)‘𝑥) ⊆ V
9		xpss12 5458	. . . 4 ⊢ (({𝑥} ⊆ (𝐵 × 𝐵) ∧ ((Hom ‘𝐶)‘𝑥) ⊆ V) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
10	7, 8, 9	sylancl 586	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 × 𝐵)) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
11		snex 5223	. . . . 5 ⊢ {𝑥} ∈ V
12		fvex 6551	. . . . 5 ⊢ ((Hom ‘𝐶)‘𝑥) ∈ V
13	11, 12	xpex 7333	. . . 4 ⊢ ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ V
14	13	elpw 4459	. . 3 ⊢ (({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ 𝒫 ((𝐵 × 𝐵) × V) ↔ ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
15	10, 14	sylibr 235	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 × 𝐵)) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ 𝒫 ((𝐵 × 𝐵) × V))
16	5, 15	fmpt3d 6743	1 ⊢ (𝜑 → 𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1522   ∈ wcel 2081  Vcvv 3437   ⊆ wss 3859  𝒫 cpw 4453  {csn 4472   × cxp 5441  ⟶wf 6221  ‘cfv 6225  Basecbs 16312  Hom chom 16405  Catccat 16764  Hom_achoma 17112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-homa 17115

This theorem is referenced by:  homarcl2  17124  homarel  17125  arwhoma  17134