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Theorem isacycgr1 32418
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 6658 . . . . 5 (𝑔 = 𝐺 → (Cycles‘𝑔) = (Cycles‘𝐺))
21breqd 5063 . . . 4 (𝑔 = 𝐺 → (𝑓(Cycles‘𝑔)𝑝𝑓(Cycles‘𝐺)𝑝))
32imbi1d 345 . . 3 (𝑔 = 𝐺 → ((𝑓(Cycles‘𝑔)𝑝𝑓 = ∅) ↔ (𝑓(Cycles‘𝐺)𝑝𝑓 = ∅)))
432albidv 1925 . 2 (𝑔 = 𝐺 → (∀𝑓𝑝(𝑓(Cycles‘𝑔)𝑝𝑓 = ∅) ↔ ∀𝑓𝑝(𝑓(Cycles‘𝐺)𝑝𝑓 = ∅)))
5 dfacycgr1 32416 . 2 AcyclicGraph = {𝑔 ∣ ∀𝑓𝑝(𝑓(Cycles‘𝑔)𝑝𝑓 = ∅)}
64, 5elab2g 3654 1 (𝐺𝑊 → (𝐺 ∈ AcyclicGraph ↔ ∀𝑓𝑝(𝑓(Cycles‘𝐺)𝑝𝑓 = ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536   = wceq 1538  wcel 2115  c0 4275   class class class wbr 5052  cfv 6343  Cyclesccycls 27570  AcyclicGraphcacycgr 32414
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-acycgr 32415
This theorem is referenced by:  acycgrcycl  32419  acycgr1v  32421
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