Theorem isacycgr1 32506

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 6645	. . . . 5 ⊢ (𝑔 = 𝐺 → (Cycles‘𝑔) = (Cycles‘𝐺))
2	1	breqd 5041	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑓(Cycles‘𝑔)𝑝 ↔ 𝑓(Cycles‘𝐺)𝑝))
3	2	imbi1d 345	. . 3 ⊢ (𝑔 = 𝐺 → ((𝑓(Cycles‘𝑔)𝑝 → 𝑓 = ∅) ↔ (𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅)))
4	3	2albidv 1924	. 2 ⊢ (𝑔 = 𝐺 → (∀𝑓∀𝑝(𝑓(Cycles‘𝑔)𝑝 → 𝑓 = ∅) ↔ ∀𝑓∀𝑝(𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅)))
5		dfacycgr1 32504	. 2 ⊢ AcyclicGraph = {𝑔 ∣ ∀𝑓∀𝑝(𝑓(Cycles‘𝑔)𝑝 → 𝑓 = ∅)}
6	4, 5	elab2g 3616	1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ AcyclicGraph ↔ ∀𝑓∀𝑝(𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅)))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ∅c0 4243   class class class wbr 5030  ‘cfv 6324  Cyclesccycls 27574  AcyclicGraphcacycgr 32502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-acycgr 32503

This theorem is referenced by:  acycgrcycl  32507  acycgr1v  32509