Theorem wlkiswwlkupgr

Description: A walk as word corresponds to a walk in a pseudograph. This variant of wlkiswwlks does not require G to be a simple pseudograph, but it requires (indirectly) the Axiom of Choice for its proof. (Contributed by Alexander van der Vekens, 21-Jul-2018) (Revised by AV, 10-Apr-2021)

	Ref	Expression
Assertion	wlkiswwlkupgr	⊢ ( 𝐺 ∈ UPGraph → ( ∃ 𝑓 𝑓 ( Walks ‘ 𝐺 ) 𝑃 ↔ 𝑃 ∈ ( WWalks ‘ 𝐺 ) ) )

Proof

Step	Hyp	Ref	Expression
1		wlkiswwlks1	⊢ ( 𝐺 ∈ UPGraph → ( 𝑓 ( Walks ‘ 𝐺 ) 𝑃 → 𝑃 ∈ ( WWalks ‘ 𝐺 ) ) )
2	1	exlimdv	⊢ ( 𝐺 ∈ UPGraph → ( ∃ 𝑓 𝑓 ( Walks ‘ 𝐺 ) 𝑃 → 𝑃 ∈ ( WWalks ‘ 𝐺 ) ) )
3		wlkiswwlksupgr2	⊢ ( 𝐺 ∈ UPGraph → ( 𝑃 ∈ ( WWalks ‘ 𝐺 ) → ∃ 𝑓 𝑓 ( Walks ‘ 𝐺 ) 𝑃 ) )
4	2 3	impbid	⊢ ( 𝐺 ∈ UPGraph → ( ∃ 𝑓 𝑓 ( Walks ‘ 𝐺 ) 𝑃 ↔ 𝑃 ∈ ( WWalks ‘ 𝐺 ) ) )