Theorem wlklnwwlklnupgr2

Description: A walk of length N as word corresponds to the sequence of vertices in a walk of length N in a pseudograph. This variant of wlklnwwlkln2 does not require G to be a simple pseudograph, but it requires (indirectly) the Axiom of Choice. (Contributed by Alexander van der Vekens, 21-Jul-2018) (Revised by AV, 12-Apr-2021)

	Ref	Expression
Assertion	wlklnwwlklnupgr2	$$⊢G\in \mathrm{UPGraph}\to \left(P\in \left(N\mathrm{WWalksN}G\right)\to \exists f\left(f\mathrm{Walks}\left(G\right)P\wedge \left|f\right|=N\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		wlkiswwlksupgr2	$$⊢G\in \mathrm{UPGraph}\to \left(P\in \mathrm{WWalks}\left(G\right)\to \exists ff\mathrm{Walks}\left(G\right)P\right)$$
2	1	wlklnwwlkln2lem	$$⊢G\in \mathrm{UPGraph}\to \left(P\in \left(N\mathrm{WWalksN}G\right)\to \exists f\left(f\mathrm{Walks}\left(G\right)P\wedge \left|f\right|=N\right)\right)$$