MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  homarcl Structured version   Visualization version   GIF version

Theorem homarcl 16659
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 3912 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → ¬ (𝑋𝐻𝑌) = ∅)
2 homarcl.h . . . . 5 𝐻 = (Homa𝐶)
3 df-homa 16657 . . . . . 6 Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
43fvmptndm 6294 . . . . 5 𝐶 ∈ Cat → (Homa𝐶) = ∅)
52, 4syl5eq 2666 . . . 4 𝐶 ∈ Cat → 𝐻 = ∅)
65oveqd 6652 . . 3 𝐶 ∈ Cat → (𝑋𝐻𝑌) = (𝑋𝑌))
7 0ov 6667 . . 3 (𝑋𝑌) = ∅
86, 7syl6eq 2670 . 2 𝐶 ∈ Cat → (𝑋𝐻𝑌) = ∅)
91, 8nsyl2 142 1 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1481  wcel 1988  c0 3907  {csn 4168  cmpt 4720   × cxp 5102  cfv 5876  (class class class)co 6635  Basecbs 15838  Hom chom 15933  Catccat 16306  Homachoma 16654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-dm 5114  df-iota 5839  df-fv 5884  df-ov 6638  df-homa 16657
This theorem is referenced by:  homarcl2  16666  homarel  16667  homa1  16668  homahom2  16669  coahom  16701  arwlid  16703  arwrid  16704  arwass  16705
  Copyright terms: Public domain W3C validator