Users' Mathboxes Mathbox for BTernaryTau < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isacycgr Structured version   Visualization version   GIF version

Theorem isacycgr 35151
Description: The property of being an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.)
Assertion
Ref Expression
isacycgr (𝐺𝑊 → (𝐺 ∈ AcyclicGraph ↔ ¬ ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝𝑓 ≠ ∅)))
Distinct variable group:   𝑓,𝐺,𝑝
Allowed substitution hints:   𝑊(𝑓,𝑝)

Proof of Theorem isacycgr
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6905 . . . . . 6 (𝑔 = 𝐺 → (Cycles‘𝑔) = (Cycles‘𝐺))
21breqd 5153 . . . . 5 (𝑔 = 𝐺 → (𝑓(Cycles‘𝑔)𝑝𝑓(Cycles‘𝐺)𝑝))
32anbi1d 631 . . . 4 (𝑔 = 𝐺 → ((𝑓(Cycles‘𝑔)𝑝𝑓 ≠ ∅) ↔ (𝑓(Cycles‘𝐺)𝑝𝑓 ≠ ∅)))
432exbidv 1923 . . 3 (𝑔 = 𝐺 → (∃𝑓𝑝(𝑓(Cycles‘𝑔)𝑝𝑓 ≠ ∅) ↔ ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝𝑓 ≠ ∅)))
54notbid 318 . 2 (𝑔 = 𝐺 → (¬ ∃𝑓𝑝(𝑓(Cycles‘𝑔)𝑝𝑓 ≠ ∅) ↔ ¬ ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝𝑓 ≠ ∅)))
6 df-acycgr 35149 . 2 AcyclicGraph = {𝑔 ∣ ¬ ∃𝑓𝑝(𝑓(Cycles‘𝑔)𝑝𝑓 ≠ ∅)}
75, 6elab2g 3679 1 (𝐺𝑊 → (𝐺 ∈ AcyclicGraph ↔ ¬ ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝𝑓 ≠ ∅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1539  wex 1778  wcel 2107  wne 2939  c0 4332   class class class wbr 5142  cfv 6560  Cyclesccycls 29806  AcyclicGraphcacycgr 35148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-acycgr 35149
This theorem is referenced by:  acycgr0v  35154  acycgr2v  35156  acycgrislfgr  35158  umgracycusgr  35160  cusgracyclt3v  35162  acycgrsubgr  35164
  Copyright terms: Public domain W3C validator