Theorem homaf 17997

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 2736	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
5	1, 2, 3, 4	homafval 17996	. 2 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥))))
6		snssi 4729	. . . . 5 ⊢ (𝑥 ∈ (𝐵 × 𝐵) → {𝑥} ⊆ (𝐵 × 𝐵))
7	6	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 × 𝐵)) → {𝑥} ⊆ (𝐵 × 𝐵))
8		ssv 3946	. . . 4 ⊢ ((Hom ‘𝐶)‘𝑥) ⊆ V
9		xpss12 5646	. . . 4 ⊢ (({𝑥} ⊆ (𝐵 × 𝐵) ∧ ((Hom ‘𝐶)‘𝑥) ⊆ V) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
10	7, 8, 9	sylancl 587	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 × 𝐵)) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
11		vsnex 5377	. . . . 5 ⊢ {𝑥} ∈ V
12		fvex 6853	. . . . 5 ⊢ ((Hom ‘𝐶)‘𝑥) ∈ V
13	11, 12	xpex 7707	. . . 4 ⊢ ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ V
14	13	elpw 4545	. . 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 7068	1 ⊢ (𝜑 → 𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ⊆ wss 3889  𝒫 cpw 4541  {csn 4567   × cxp 5629  ⟶wf 6494  ‘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:  homarcl2  18002  homarel  18003  arwhoma  18012