Theorem isacycgr1 35677

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 6885	. . . . 5 ⊢ (𝑔 = 𝐺 → (Cycles‘𝑔) = (Cycles‘𝐺))
2	1	breqd 5122	. . . 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 1956	. 2 ⊢ (𝑔 = 𝐺 → (∀𝑓∀𝑝(𝑓(Cycles‘𝑔)𝑝 → 𝑓 = ∅) ↔ ∀𝑓∀𝑝(𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅)))
5		dfacycgr1 35675	. 2 ⊢ AcyclicGraph = {𝑔 ∣ ∀𝑓∀𝑝(𝑓(Cycles‘𝑔)𝑝 → 𝑓 = ∅)}
6	4, 5	elab2g 3641	1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ AcyclicGraph ↔ ∀𝑓∀𝑝(𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570   ∈ wcel 2146  ∅c0 4286   class class class wbr 5111  ‘cfv 6540  Cyclesccycls 30201  AcyclicGraphcacycgr 35673

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 2737

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 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-acycgr 35674

This theorem is used by:  acycgrcycl  35678  acycgr1v  35680