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Theorem isacycgr 34136
Description: The property of being an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.)
Assertion
Ref Expression
isacycgr (𝐺 ∈ π‘Š β†’ (𝐺 ∈ AcyclicGraph ↔ Β¬ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ 𝑓 β‰  βˆ…)))
Distinct variable group:   𝑓,𝐺,𝑝
Allowed substitution hints:   π‘Š(𝑓,𝑝)

Proof of Theorem isacycgr
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6892 . . . . . 6 (𝑔 = 𝐺 β†’ (Cyclesβ€˜π‘”) = (Cyclesβ€˜πΊ))
21breqd 5160 . . . . 5 (𝑔 = 𝐺 β†’ (𝑓(Cyclesβ€˜π‘”)𝑝 ↔ 𝑓(Cyclesβ€˜πΊ)𝑝))
32anbi1d 631 . . . 4 (𝑔 = 𝐺 β†’ ((𝑓(Cyclesβ€˜π‘”)𝑝 ∧ 𝑓 β‰  βˆ…) ↔ (𝑓(Cyclesβ€˜πΊ)𝑝 ∧ 𝑓 β‰  βˆ…)))
432exbidv 1928 . . 3 (𝑔 = 𝐺 β†’ (βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜π‘”)𝑝 ∧ 𝑓 β‰  βˆ…) ↔ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ 𝑓 β‰  βˆ…)))
54notbid 318 . 2 (𝑔 = 𝐺 β†’ (Β¬ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜π‘”)𝑝 ∧ 𝑓 β‰  βˆ…) ↔ Β¬ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ 𝑓 β‰  βˆ…)))
6 df-acycgr 34134 . 2 AcyclicGraph = {𝑔 ∣ Β¬ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜π‘”)𝑝 ∧ 𝑓 β‰  βˆ…)}
75, 6elab2g 3671 1 (𝐺 ∈ π‘Š β†’ (𝐺 ∈ AcyclicGraph ↔ Β¬ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ 𝑓 β‰  βˆ…)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   β‰  wne 2941  βˆ…c0 4323   class class class wbr 5149  β€˜cfv 6544  Cyclesccycls 29042  AcyclicGraphcacycgr 34133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-acycgr 34134
This theorem is referenced by:  acycgr0v  34139  acycgr2v  34141  acycgrislfgr  34143  umgracycusgr  34145  cusgracyclt3v  34147  acycgrsubgr  34149
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