Theorem isacycgr

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

	Ref	Expression
Assertion	isacycgr	⊢ ( 𝐺 ∈ 𝑊 → ( 𝐺 ∈ AcyclicGraph ↔ ¬ ∃ 𝑓 ∃ 𝑝 ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 ∧ 𝑓 ≠ ∅ ) ) )

Proof

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