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Theorem homaf 17119
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 2795 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
51, 2, 3, 4homafval 17118 . 2 (𝜑𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥))))
6 snssi 4648 . . . . 5 (𝑥 ∈ (𝐵 × 𝐵) → {𝑥} ⊆ (𝐵 × 𝐵))
76adantl 482 . . . 4 ((𝜑𝑥 ∈ (𝐵 × 𝐵)) → {𝑥} ⊆ (𝐵 × 𝐵))
8 ssv 3912 . . . 4 ((Hom ‘𝐶)‘𝑥) ⊆ V
9 xpss12 5458 . . . 4 (({𝑥} ⊆ (𝐵 × 𝐵) ∧ ((Hom ‘𝐶)‘𝑥) ⊆ V) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
107, 8, 9sylancl 586 . . 3 ((𝜑𝑥 ∈ (𝐵 × 𝐵)) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
11 snex 5223 . . . . 5 {𝑥} ∈ V
12 fvex 6551 . . . . 5 ((Hom ‘𝐶)‘𝑥) ∈ V
1311, 12xpex 7333 . . . 4 ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ V
1413elpw 4459 . . 3 (({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ 𝒫 ((𝐵 × 𝐵) × V) ↔ ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
1510, 14sylibr 235 . 2 ((𝜑𝑥 ∈ (𝐵 × 𝐵)) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ 𝒫 ((𝐵 × 𝐵) × V))
165, 15fmpt3d 6743 1 (𝜑𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wcel 2081  Vcvv 3437  wss 3859  𝒫 cpw 4453  {csn 4472   × cxp 5441  wf 6221  cfv 6225  Basecbs 16312  Hom chom 16405  Catccat 16764  Homachoma 17112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-homa 17115
This theorem is referenced by:  homarcl2  17124  homarel  17125  arwhoma  17134
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