Theorem wlklnwwlkn

Description: A walk of length N as word corresponds to a walk with length N in a simple pseudograph. (Contributed by Alexander van der Vekens, 21-Jul-2018) (Revised by AV, 12-Apr-2021)

	Ref	Expression
Assertion	wlklnwwlkn	⊢ ( 𝐺 ∈ USPGraph → ( ∃ 𝑓 ( 𝑓 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( ♯ ‘ 𝑓 ) = 𝑁 ) ↔ 𝑃 ∈ ( 𝑁 WWalksN 𝐺 ) ) )

Proof

Step	Hyp	Ref	Expression
1		uspgrupgr	⊢ ( 𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph )
2		wlklnwwlkln1	⊢ ( 𝐺 ∈ UPGraph → ( ( 𝑓 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( ♯ ‘ 𝑓 ) = 𝑁 ) → 𝑃 ∈ ( 𝑁 WWalksN 𝐺 ) ) )
3	1 2	syl	⊢ ( 𝐺 ∈ USPGraph → ( ( 𝑓 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( ♯ ‘ 𝑓 ) = 𝑁 ) → 𝑃 ∈ ( 𝑁 WWalksN 𝐺 ) ) )
4	3	exlimdv	⊢ ( 𝐺 ∈ USPGraph → ( ∃ 𝑓 ( 𝑓 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( ♯ ‘ 𝑓 ) = 𝑁 ) → 𝑃 ∈ ( 𝑁 WWalksN 𝐺 ) ) )
5		wlklnwwlkln2	⊢ ( 𝐺 ∈ USPGraph → ( 𝑃 ∈ ( 𝑁 WWalksN 𝐺 ) → ∃ 𝑓 ( 𝑓 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( ♯ ‘ 𝑓 ) = 𝑁 ) ) )
6	4 5	impbid	⊢ ( 𝐺 ∈ USPGraph → ( ∃ 𝑓 ( 𝑓 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( ♯ ‘ 𝑓 ) = 𝑁 ) ↔ 𝑃 ∈ ( 𝑁 WWalksN 𝐺 ) ) )