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Theorem isacycgr 32843
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 6736 . . . . . 6 (𝑔 = 𝐺 → (Cycles‘𝑔) = (Cycles‘𝐺))
21breqd 5079 . . . . 5 (𝑔 = 𝐺 → (𝑓(Cycles‘𝑔)𝑝𝑓(Cycles‘𝐺)𝑝))
32anbi1d 633 . . . 4 (𝑔 = 𝐺 → ((𝑓(Cycles‘𝑔)𝑝𝑓 ≠ ∅) ↔ (𝑓(Cycles‘𝐺)𝑝𝑓 ≠ ∅)))
432exbidv 1932 . . 3 (𝑔 = 𝐺 → (∃𝑓𝑝(𝑓(Cycles‘𝑔)𝑝𝑓 ≠ ∅) ↔ ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝𝑓 ≠ ∅)))
54notbid 321 . 2 (𝑔 = 𝐺 → (¬ ∃𝑓𝑝(𝑓(Cycles‘𝑔)𝑝𝑓 ≠ ∅) ↔ ¬ ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝𝑓 ≠ ∅)))
6 df-acycgr 32841 . 2 AcyclicGraph = {𝑔 ∣ ¬ ∃𝑓𝑝(𝑓(Cycles‘𝑔)𝑝𝑓 ≠ ∅)}
75, 6elab2g 3602 1 (𝐺𝑊 → (𝐺 ∈ AcyclicGraph ↔ ¬ ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝𝑓 ≠ ∅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1543  wex 1787  wcel 2111  wne 2941  c0 4252   class class class wbr 5068  cfv 6398  Cyclesccycls 27896  AcyclicGraphcacycgr 32840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3071  df-v 3423  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4835  df-br 5069  df-iota 6356  df-fv 6406  df-acycgr 32841
This theorem is referenced by:  acycgr0v  32846  acycgr2v  32848  acycgrislfgr  32850  umgracycusgr  32852  cusgracyclt3v  32854  acycgrsubgr  32856
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