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Theorem homaf 16451
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 4279 . . . . . 6 (𝑥 ∈ (𝐵 × 𝐵) → {𝑥} ⊆ (𝐵 × 𝐵))
21adantl 480 . . . . 5 ((𝜑𝑥 ∈ (𝐵 × 𝐵)) → {𝑥} ⊆ (𝐵 × 𝐵))
3 ssv 3587 . . . . 5 ((Hom ‘𝐶)‘𝑥) ⊆ V
4 xpss12 5136 . . . . 5 (({𝑥} ⊆ (𝐵 × 𝐵) ∧ ((Hom ‘𝐶)‘𝑥) ⊆ V) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
52, 3, 4sylancl 692 . . . 4 ((𝜑𝑥 ∈ (𝐵 × 𝐵)) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
6 snex 4829 . . . . . 6 {𝑥} ∈ V
7 fvex 6097 . . . . . 6 ((Hom ‘𝐶)‘𝑥) ∈ V
86, 7xpex 6837 . . . . 5 ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ V
98elpw 4113 . . . 4 (({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ 𝒫 ((𝐵 × 𝐵) × V) ↔ ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V))
105, 9sylibr 222 . . 3 ((𝜑𝑥 ∈ (𝐵 × 𝐵)) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ 𝒫 ((𝐵 × 𝐵) × V))
11 eqid 2609 . . 3 (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥))) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥)))
1210, 11fmptd 6276 . 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 2609 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
1713, 14, 15, 16homafval 16450 . . 3 (𝜑𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥))))
1817feq1d 5928 . 2 (𝜑 → (𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V) ↔ (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥))):(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V)))
1912, 18mpbird 245 1 (𝜑𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  Vcvv 3172  wss 3539  𝒫 cpw 4107  {csn 4124  cmpt 4637   × cxp 5025  wf 5785  cfv 5789  Basecbs 15643  Hom chom 15727  Catccat 16096  Homachoma 16444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-homa 16447
This theorem is referenced by:  homarcl2  16456  homarel  16457  arwhoma  16466
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