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Theorem homaf 16886
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 snssi 4472 . . . . . 6 (𝑥 ∈ (𝐵 × 𝐵) → {𝑥} ⊆ (𝐵 × 𝐵))
21adantl 467 . . . . 5 ((𝜑𝑥 ∈ (𝐵 × 𝐵)) → {𝑥} ⊆ (𝐵 × 𝐵))
3 ssv 3772 . . . . 5 ((Hom ‘𝐶)‘𝑥) ⊆ V
4 xpss12 5264 . . . . 5 (({𝑥} ⊆ (𝐵 × 𝐵) ∧ ((Hom ‘𝐶)‘𝑥) ⊆ V) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
52, 3, 4sylancl 566 . . . 4 ((𝜑𝑥 ∈ (𝐵 × 𝐵)) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
6 snex 5036 . . . . . 6 {𝑥} ∈ V
7 fvex 6342 . . . . . 6 ((Hom ‘𝐶)‘𝑥) ∈ V
86, 7xpex 7108 . . . . 5 ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ V
98elpw 4301 . . . 4 (({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ 𝒫 ((𝐵 × 𝐵) × V) ↔ ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
105, 9sylibr 224 . . 3 ((𝜑𝑥 ∈ (𝐵 × 𝐵)) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ 𝒫 ((𝐵 × 𝐵) × V))
11 eqid 2770 . . 3 (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥))) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥)))
1210, 11fmptd 6527 . 2 (𝜑 → (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥))):(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))
13 homarcl.h . . . 4 𝐻 = (Homa𝐶)
14 homafval.b . . . 4 𝐵 = (Base‘𝐶)
15 homafval.c . . . 4 (𝜑𝐶 ∈ Cat)
16 eqid 2770 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
1713, 14, 15, 16homafval 16885 . . 3 (𝜑𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥))))
1817feq1d 6170 . 2 (𝜑 → (𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V) ↔ (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥))):(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V)))
1912, 18mpbird 247 1 (𝜑𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  Vcvv 3349  wss 3721  𝒫 cpw 4295  {csn 4314  cmpt 4861   × cxp 5247  wf 6027  cfv 6031  Basecbs 16063  Hom chom 16159  Catccat 16531  Homachoma 16879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-homa 16882
This theorem is referenced by:  homarcl2  16891  homarel  16892  arwhoma  16901
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