Theorem isacycgr 35658

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

Assertion

Ref	Expression
isacycgr	⊢ (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 isacycgr

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6888	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Cycles‘𝑔) = (Cycles‘𝐺))
2	1	breqd 5125	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑓(Cycles‘𝑔)𝑝 ↔ 𝑓(Cycles‘𝐺)𝑝))
3	2	anbi1d 643	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅) ↔ (𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)))
4	3	2exbidv 1957	. . 3 ⊢ (𝑔 = 𝐺 → (∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅) ↔ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)))
5	4	notbid 321	. 2 ⊢ (𝑔 = 𝐺 → (¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅) ↔ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)))
6		df-acycgr 35656	. 2 ⊢ AcyclicGraph = {𝑔 ∣ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅)}
7	5, 6	elab2g 3642	1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ AcyclicGraph ↔ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146   ≠ wne 2961  ∅c0 4289   class class class wbr 5114  ‘cfv 6543  Cyclesccycls 30171  AcyclicGraphcacycgr 35655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6499  df-fv 6551  df-acycgr 35656

This theorem is used by:  acycgr0v  35661  acycgr2v  35663  acycgrislfgr  35665  umgracycusgr  35667  cusgracyclt3v  35669  acycgrsubgr  35671