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Theorem homarcl 17074
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 4148 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → ¬ (𝑋𝐻𝑌) = ∅)
2 homarcl.h . . . . 5 𝐻 = (Homa𝐶)
3 df-homa 17072 . . . . . 6 Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
43fvmptndm 6572 . . . . 5 𝐶 ∈ Cat → (Homa𝐶) = ∅)
52, 4syl5eq 2826 . . . 4 𝐶 ∈ Cat → 𝐻 = ∅)
65oveqd 6941 . . 3 𝐶 ∈ Cat → (𝑋𝐻𝑌) = (𝑋𝑌))
7 0ov 6960 . . 3 (𝑋𝑌) = ∅
86, 7syl6eq 2830 . 2 𝐶 ∈ Cat → (𝑋𝐻𝑌) = ∅)
91, 8nsyl2 145 1 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1601  wcel 2107  c0 4141  {csn 4398  cmpt 4967   × cxp 5355  cfv 6137  (class class class)co 6924  Basecbs 16266  Hom chom 16360  Catccat 16721  Homachoma 17069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-mpt 4968  df-dm 5367  df-iota 6101  df-fv 6145  df-ov 6927  df-homa 17072
This theorem is referenced by:  homarcl2  17081  homarel  17082  homa1  17083  homahom2  17084  coahom  17116  arwlid  17118  arwrid  17119  arwass  17120
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