Theorem homaf 16991

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 4526	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 × 𝐵) → {𝑥} ⊆ (𝐵 × 𝐵))
2	1	adantl 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 × 𝐵)) → {𝑥} ⊆ (𝐵 × 𝐵))
3		ssv 3820	. . . . 5 ⊢ ((Hom ‘𝐶)‘𝑥) ⊆ V
4		xpss12 5326	. . . . 5 ⊢ (({𝑥} ⊆ (𝐵 × 𝐵) ∧ ((Hom ‘𝐶)‘𝑥) ⊆ V) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
5	2, 3, 4	sylancl 581	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 × 𝐵)) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
6		snex 5098	. . . . . 6 ⊢ {𝑥} ∈ V
7		fvex 6423	. . . . . 6 ⊢ ((Hom ‘𝐶)‘𝑥) ∈ V
8	6, 7	xpex 7195	. . . . 5 ⊢ ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ V
9	8	elpw 4354	. . . 4 ⊢ (({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ 𝒫 ((𝐵 × 𝐵) × V) ↔ ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
10	5, 9	sylibr 226	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 × 𝐵)) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ 𝒫 ((𝐵 × 𝐵) × V))
11	10	fmpttd 6610	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥))):(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))
12		homarcl.h	. . . 4 ⊢ 𝐻 = (Homa‘𝐶)
13		homafval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
14		homafval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
15		eqid 2798	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
16	12, 13, 14, 15	homafval 16990	. . 3 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥))))
17	16	feq1d 6240	. 2 ⊢ (𝜑 → (𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V) ↔ (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥))):(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V)))
18	11, 17	mpbird 249	1 ⊢ (𝜑 → 𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))

Syntax hints:   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157  Vcvv 3384   ⊆ wss 3768  𝒫 cpw 4348  {csn 4367   ↦ cmpt 4921   × cxp 5309  ⟶wf 6096  ‘cfv 6100  Basecbs 16181  Hom chom 16275  Catccat 16636  Hom_achoma 16984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-rep 4963  ax-sep 4974  ax-nul 4982  ax-pow 5034  ax-pr 5096  ax-un 7182

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ne 2971  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3386  df-sbc 3633  df-csb 3728  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4628  df-iun 4711  df-br 4843  df-opab 4905  df-mpt 4922  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6063  df-fun 6102  df-fn 6103  df-f 6104  df-f1 6105  df-fo 6106  df-f1o 6107  df-fv 6108  df-homa 16987

This theorem is referenced by:  homarcl2  16996  homarel  16997  arwhoma  17006