Theorem isacycgr 35130

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 6907	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Cycles‘𝑔) = (Cycles‘𝐺))
2	1	breqd 5159	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑓(Cycles‘𝑔)𝑝 ↔ 𝑓(Cycles‘𝐺)𝑝))
3	2	anbi1d 631	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅) ↔ (𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)))
4	3	2exbidv 1922	. . 3 ⊢ (𝑔 = 𝐺 → (∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅) ↔ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)))
5	4	notbid 318	. 2 ⊢ (𝑔 = 𝐺 → (¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅) ↔ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)))
6		df-acycgr 35128	. 2 ⊢ AcyclicGraph = {𝑔 ∣ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅)}
7	5, 6	elab2g 3683	1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ AcyclicGraph ↔ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1776   ∈ wcel 2106   ≠ wne 2938  ∅c0 4339   class class class wbr 5148  ‘cfv 6563  Cyclesccycls 29818  AcyclicGraphcacycgr 35127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-acycgr 35128

This theorem is referenced by:  acycgr0v  35133  acycgr2v  35135  acycgrislfgr  35137  umgracycusgr  35139  cusgracyclt3v  35141  acycgrsubgr  35143