Theorem isacycgr1 32414

Description: The property of being an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.)

Assertion

Ref	Expression
isacycgr1	⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ AcyclicGraph ↔ ∀𝑓∀𝑝(𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅)))

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.

Step	Hyp	Ref	Expression
1		fveq2 6663	. . . . 5 ⊢ (𝑔 = 𝐺 → (Cycles‘𝑔) = (Cycles‘𝐺))
2	1	breqd 5070	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑓(Cycles‘𝑔)𝑝 ↔ 𝑓(Cycles‘𝐺)𝑝))
3	2	imbi1d 344	. . 3 ⊢ (𝑔 = 𝐺 → ((𝑓(Cycles‘𝑔)𝑝 → 𝑓 = ∅) ↔ (𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅)))
4	3	2albidv 1923	. 2 ⊢ (𝑔 = 𝐺 → (∀𝑓∀𝑝(𝑓(Cycles‘𝑔)𝑝 → 𝑓 = ∅) ↔ ∀𝑓∀𝑝(𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅)))
5		dfacycgr1 32412	. 2 ⊢ AcyclicGraph = {𝑔 ∣ ∀𝑓∀𝑝(𝑓(Cycles‘𝑔)𝑝 → 𝑓 = ∅)}
6	4, 5	elab2g 3664	1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ AcyclicGraph ↔ ∀𝑓∀𝑝(𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅)))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1534   = wceq 1536   ∈ wcel 2113  ∅c0 4284   class class class wbr 5059  ‘cfv 6348  Cyclesccycls 27564  AcyclicGraphcacycgr 32410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-rab 3146  df-v 3493  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5060  df-iota 6307  df-fv 6356  df-acycgr 32411

This theorem is referenced by:  acycgrcycl  32415  acycgr1v  32417