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Theorem homaf 17536
Description: Functionality of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
homarcl.h 𝐻 = (Homa𝐶)
homafval.b 𝐵 = (Base‘𝐶)
homafval.c (𝜑𝐶 ∈ Cat)
Assertion
Ref Expression
homaf (𝜑𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))

Proof of Theorem homaf
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 homarcl.h . . 3 𝐻 = (Homa𝐶)
2 homafval.b . . 3 𝐵 = (Base‘𝐶)
3 homafval.c . . 3 (𝜑𝐶 ∈ Cat)
4 eqid 2737 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
51, 2, 3, 4homafval 17535 . 2 (𝜑𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥))))
6 snssi 4721 . . . . 5 (𝑥 ∈ (𝐵 × 𝐵) → {𝑥} ⊆ (𝐵 × 𝐵))
76adantl 485 . . . 4 ((𝜑𝑥 ∈ (𝐵 × 𝐵)) → {𝑥} ⊆ (𝐵 × 𝐵))
8 ssv 3925 . . . 4 ((Hom ‘𝐶)‘𝑥) ⊆ V
9 xpss12 5566 . . . 4 (({𝑥} ⊆ (𝐵 × 𝐵) ∧ ((Hom ‘𝐶)‘𝑥) ⊆ V) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
107, 8, 9sylancl 589 . . 3 ((𝜑𝑥 ∈ (𝐵 × 𝐵)) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
11 snex 5324 . . . . 5 {𝑥} ∈ V
12 fvex 6730 . . . . 5 ((Hom ‘𝐶)‘𝑥) ∈ V
1311, 12xpex 7538 . . . 4 ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ V
1413elpw 4517 . . 3 (({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ 𝒫 ((𝐵 × 𝐵) × V) ↔ ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
1510, 14sylibr 237 . 2 ((𝜑𝑥 ∈ (𝐵 × 𝐵)) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ 𝒫 ((𝐵 × 𝐵) × V))
165, 15fmpt3d 6933 1 (𝜑𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  Vcvv 3408  wss 3866  𝒫 cpw 4513  {csn 4541   × cxp 5549  wf 6376  cfv 6380  Basecbs 16760  Hom chom 16813  Catccat 17167  Homachoma 17529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-homa 17532
This theorem is referenced by:  homarcl2  17541  homarel  17542  arwhoma  17551
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