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

Theorem homarcl 18073
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 4340 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → ¬ (𝑋𝐻𝑌) = ∅)
2 homarcl.h . . . . 5 𝐻 = (Homa𝐶)
3 df-homa 18071 . . . . . 6 Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
43fvmptndm 7047 . . . . 5 𝐶 ∈ Cat → (Homa𝐶) = ∅)
52, 4eqtrid 2789 . . . 4 𝐶 ∈ Cat → 𝐻 = ∅)
65oveqd 7448 . . 3 𝐶 ∈ Cat → (𝑋𝐻𝑌) = (𝑋𝑌))
7 0ov 7468 . . 3 (𝑋𝑌) = ∅
86, 7eqtrdi 2793 . 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 1540  wcel 2108  c0 4333  {csn 4626  cmpt 5225   × cxp 5683  cfv 6561  (class class class)co 7431  Basecbs 17247  Hom chom 17308  Catccat 17707  Homachoma 18068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-dm 5695  df-iota 6514  df-fv 6569  df-ov 7434  df-homa 18071
This theorem is referenced by:  homarcl2  18080  homarel  18081  homa1  18082  homahom2  18083  coahom  18115  arwlid  18117  arwrid  18118  arwass  18119
  Copyright terms: Public domain W3C validator