Theorem wwlksnonfi

Description: In a finite graph, the set of walks of a fixed length between two vertices is finite. (Contributed by Alexander van der Vekens, 4-Mar-2018) (Revised by AV, 15-May-2021) (Proof shortened by AV, 15-Mar-2022)

	Ref	Expression
Assertion	wwlksnonfi	|- ( ( Vtx ` G ) e. Fin -> ( A ( N WWalksNOn G ) B ) e. Fin )

Proof

Step	Hyp	Ref	Expression
1		eqid	|- ( Vtx ` G ) = ( Vtx ` G )
2	1	iswwlksnon	|- ( A ( N WWalksNOn G ) B ) = { w e. ( N WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` N ) = B ) }
3		wwlksnfi	|- ( ( Vtx ` G ) e. Fin -> ( N WWalksN G ) e. Fin )
4		rabfi	|- ( ( N WWalksN G ) e. Fin -> { w e. ( N WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` N ) = B ) } e. Fin )
5	3 4	syl	|- ( ( Vtx ` G ) e. Fin -> { w e. ( N WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` N ) = B ) } e. Fin )
6	2 5	eqeltrid	|- ( ( Vtx ` G ) e. Fin -> ( A ( N WWalksNOn G ) B ) e. Fin )