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Theorem isacycgr1 34797
Description: The property of being an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.)
Assertion
Ref Expression
isacycgr1 (𝐺 ∈ π‘Š β†’ (𝐺 ∈ AcyclicGraph ↔ βˆ€π‘“βˆ€π‘(𝑓(Cyclesβ€˜πΊ)𝑝 β†’ 𝑓 = βˆ…)))
Distinct variable group:   𝑓,𝐺,𝑝
Allowed substitution hints:   π‘Š(𝑓,𝑝)

Proof of Theorem isacycgr1
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6902 . . . . 5 (𝑔 = 𝐺 β†’ (Cyclesβ€˜π‘”) = (Cyclesβ€˜πΊ))
21breqd 5163 . . . 4 (𝑔 = 𝐺 β†’ (𝑓(Cyclesβ€˜π‘”)𝑝 ↔ 𝑓(Cyclesβ€˜πΊ)𝑝))
32imbi1d 340 . . 3 (𝑔 = 𝐺 β†’ ((𝑓(Cyclesβ€˜π‘”)𝑝 β†’ 𝑓 = βˆ…) ↔ (𝑓(Cyclesβ€˜πΊ)𝑝 β†’ 𝑓 = βˆ…)))
432albidv 1918 . 2 (𝑔 = 𝐺 β†’ (βˆ€π‘“βˆ€π‘(𝑓(Cyclesβ€˜π‘”)𝑝 β†’ 𝑓 = βˆ…) ↔ βˆ€π‘“βˆ€π‘(𝑓(Cyclesβ€˜πΊ)𝑝 β†’ 𝑓 = βˆ…)))
5 dfacycgr1 34795 . 2 AcyclicGraph = {𝑔 ∣ βˆ€π‘“βˆ€π‘(𝑓(Cyclesβ€˜π‘”)𝑝 β†’ 𝑓 = βˆ…)}
64, 5elab2g 3671 1 (𝐺 ∈ π‘Š β†’ (𝐺 ∈ AcyclicGraph ↔ βˆ€π‘“βˆ€π‘(𝑓(Cyclesβ€˜πΊ)𝑝 β†’ 𝑓 = βˆ…)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205  βˆ€wal 1531   = wceq 1533   ∈ wcel 2098  βˆ…c0 4326   class class class wbr 5152  β€˜cfv 6553  Cyclesccycls 29627  AcyclicGraphcacycgr 34793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-acycgr 34794
This theorem is referenced by:  acycgrcycl  34798  acycgr1v  34800
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