Theorem homaf 17932

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 2731	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
5	1, 2, 3, 4	homafval 17931	. 2 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥))))
6		snssi 4755	. . . . 5 ⊢ (𝑥 ∈ (𝐵 × 𝐵) → {𝑥} ⊆ (𝐵 × 𝐵))
7	6	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 × 𝐵)) → {𝑥} ⊆ (𝐵 × 𝐵))
8		ssv 3954	. . . 4 ⊢ ((Hom ‘𝐶)‘𝑥) ⊆ V
9		xpss12 5626	. . . 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		vsnex 5367	. . . . 5 ⊢ {𝑥} ∈ V
12		fvex 6830	. . . . 5 ⊢ ((Hom ‘𝐶)‘𝑥) ∈ V
13	11, 12	xpex 7681	. . . 4 ⊢ ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ V
14	13	elpw 4549	. . 3 ⊢ (({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ 𝒫 ((𝐵 × 𝐵) × V) ↔ ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
15	10, 14	sylibr 234	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 × 𝐵)) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ 𝒫 ((𝐵 × 𝐵) × V))
16	5, 15	fmpt3d 7044	1 ⊢ (𝜑 → 𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ⊆ wss 3897  𝒫 cpw 4545  {csn 4571   × cxp 5609  ⟶wf 6472  ‘cfv 6476  Basecbs 17115  Hom chom 17167  Catccat 17565  Hom_achoma 17925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-homa 17928

This theorem is referenced by:  homarcl2  17937  homarel  17938  arwhoma  17947