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Theorem isacycgr1 33552
Description: The property of being an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.)
Assertion
Ref Expression
isacycgr1 (𝐺𝑊 → (𝐺 ∈ AcyclicGraph ↔ ∀𝑓𝑝(𝑓(Cycles‘𝐺)𝑝𝑓 = ∅)))
Distinct variable group:   𝑓,𝐺,𝑝
Allowed substitution hints:   𝑊(𝑓,𝑝)

Proof of Theorem isacycgr1
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6839 . . . . 5 (𝑔 = 𝐺 → (Cycles‘𝑔) = (Cycles‘𝐺))
21breqd 5114 . . . 4 (𝑔 = 𝐺 → (𝑓(Cycles‘𝑔)𝑝𝑓(Cycles‘𝐺)𝑝))
32imbi1d 341 . . 3 (𝑔 = 𝐺 → ((𝑓(Cycles‘𝑔)𝑝𝑓 = ∅) ↔ (𝑓(Cycles‘𝐺)𝑝𝑓 = ∅)))
432albidv 1926 . 2 (𝑔 = 𝐺 → (∀𝑓𝑝(𝑓(Cycles‘𝑔)𝑝𝑓 = ∅) ↔ ∀𝑓𝑝(𝑓(Cycles‘𝐺)𝑝𝑓 = ∅)))
5 dfacycgr1 33550 . 2 AcyclicGraph = {𝑔 ∣ ∀𝑓𝑝(𝑓(Cycles‘𝑔)𝑝𝑓 = ∅)}
64, 5elab2g 3630 1 (𝐺𝑊 → (𝐺 ∈ AcyclicGraph ↔ ∀𝑓𝑝(𝑓(Cycles‘𝐺)𝑝𝑓 = ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1539   = wceq 1541  wcel 2106  c0 4280   class class class wbr 5103  cfv 6493  Cyclesccycls 28562  AcyclicGraphcacycgr 33548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-iota 6445  df-fv 6501  df-acycgr 33549
This theorem is referenced by:  acycgrcycl  33553  acycgr1v  33555
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