Theorem wspthnonfi

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

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

Proof

Step	Hyp	Ref	Expression
1		wwlksnonfi	|- ( ( Vtx ` G ) e. Fin -> ( A ( N WWalksNOn G ) B ) e. Fin )
2		wspthsswwlknon	|- ( A ( N WSPathsNOn G ) B ) C_ ( A ( N WWalksNOn G ) B )
3	2	a1i	|- ( ( Vtx ` G ) e. Fin -> ( A ( N WSPathsNOn G ) B ) C_ ( A ( N WWalksNOn G ) B ) )
4	1 3	ssfid	|- ( ( Vtx ` G ) e. Fin -> ( A ( N WSPathsNOn G ) B ) e. Fin )