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Theorem homarcl 18082
Description: Reverse closure for an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypothesis
Ref Expression
homarcl.h 𝐻 = (Homa𝐶)
Assertion
Ref Expression
homarcl (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)

Proof of Theorem homarcl
Dummy variables 𝑥 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0i 4346 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → ¬ (𝑋𝐻𝑌) = ∅)
2 homarcl.h . . . . 5 𝐻 = (Homa𝐶)
3 df-homa 18080 . . . . . 6 Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
43fvmptndm 7047 . . . . 5 𝐶 ∈ Cat → (Homa𝐶) = ∅)
52, 4eqtrid 2787 . . . 4 𝐶 ∈ Cat → 𝐻 = ∅)
65oveqd 7448 . . 3 𝐶 ∈ Cat → (𝑋𝐻𝑌) = (𝑋𝑌))
7 0ov 7468 . . 3 (𝑋𝑌) = ∅
86, 7eqtrdi 2791 . 2 𝐶 ∈ Cat → (𝑋𝐻𝑌) = ∅)
91, 8nsyl2 141 1 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wcel 2106  c0 4339  {csn 4631  cmpt 5231   × cxp 5687  cfv 6563  (class class class)co 7431  Basecbs 17245  Hom chom 17309  Catccat 17709  Homachoma 18077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-dm 5699  df-iota 6516  df-fv 6571  df-ov 7434  df-homa 18080
This theorem is referenced by:  homarcl2  18089  homarel  18090  homa1  18091  homahom2  18092  coahom  18124  arwlid  18126  arwrid  18127  arwass  18128
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