Theorem homaf 16503

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		snssi 4280	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 × 𝐵) → {𝑥} ⊆ (𝐵 × 𝐵))
2	1	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 × 𝐵)) → {𝑥} ⊆ (𝐵 × 𝐵))
3		ssv 3588	. . . . 5 ⊢ ((Hom ‘𝐶)‘𝑥) ⊆ V
4		xpss12 5148	. . . . 5 ⊢ (({𝑥} ⊆ (𝐵 × 𝐵) ∧ ((Hom ‘𝐶)‘𝑥) ⊆ V) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
5	2, 3, 4	sylancl 693	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 × 𝐵)) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
6		snex 4835	. . . . . 6 ⊢ {𝑥} ∈ V
7		fvex 6113	. . . . . 6 ⊢ ((Hom ‘𝐶)‘𝑥) ∈ V
8	6, 7	xpex 6860	. . . . 5 ⊢ ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ V
9	8	elpw 4114	. . . 4 ⊢ (({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ 𝒫 ((𝐵 × 𝐵) × V) ↔ ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
10	5, 9	sylibr 223	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 × 𝐵)) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ 𝒫 ((𝐵 × 𝐵) × V))
11		eqid 2610	. . 3 ⊢ (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥))) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥)))
12	10, 11	fmptd 6292	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥))):(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))
13		homarcl.h	. . . 4 ⊢ 𝐻 = (Homa‘𝐶)
14		homafval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
15		homafval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
16		eqid 2610	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
17	13, 14, 15, 16	homafval 16502	. . 3 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥))))
18	17	feq1d 5943	. 2 ⊢ (𝜑 → (𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V) ↔ (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥))):(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V)))
19	12, 18	mpbird 246	1 ⊢ (𝜑 → 𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125   ↦ cmpt 4643   × cxp 5036  ⟶wf 5800  ‘cfv 5804  Basecbs 15695  Hom chom 15779  Catccat 16148  Hom_achoma 16496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-homa 16499

This theorem is referenced by:  homarcl2  16508  homarel  16509  arwhoma  16518