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

Theorem homarcl 17276
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 4296 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → ¬ (𝑋𝐻𝑌) = ∅)
2 homarcl.h . . . . 5 𝐻 = (Homa𝐶)
3 df-homa 17274 . . . . . 6 Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
43fvmptndm 6790 . . . . 5 𝐶 ∈ Cat → (Homa𝐶) = ∅)
52, 4syl5eq 2865 . . . 4 𝐶 ∈ Cat → 𝐻 = ∅)
65oveqd 7162 . . 3 𝐶 ∈ Cat → (𝑋𝐻𝑌) = (𝑋𝑌))
7 0ov 7182 . . 3 (𝑋𝑌) = ∅
86, 7syl6eq 2869 . 2 𝐶 ∈ Cat → (𝑋𝐻𝑌) = ∅)
91, 8nsyl2 143 1 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1528  wcel 2105  c0 4288  {csn 4557  cmpt 5137   × cxp 5546  cfv 6348  (class class class)co 7145  Basecbs 16471  Hom chom 16564  Catccat 16923  Homachoma 17271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-dm 5558  df-iota 6307  df-fv 6356  df-ov 7148  df-homa 17274
This theorem is referenced by:  homarcl2  17283  homarel  17284  homa1  17285  homahom2  17286  coahom  17318  arwlid  17320  arwrid  17321  arwass  17322
  Copyright terms: Public domain W3C validator