Theorem wspthnfi

Description: In a finite graph, the set of simple paths of a fixed length is finite. (Contributed by Alexander van der Vekens, 4-Mar-2018) (Revised by AV, 18-May-2021)

	Ref	Expression
Assertion	wspthnfi	|- ( ( Vtx ` G ) e. Fin -> ( N WSPathsN G ) e. Fin )

Proof

Step	Hyp	Ref	Expression
1		wwlksnfi	|- ( ( Vtx ` G ) e. Fin -> ( N WWalksN G ) e. Fin )
2		wspthsswwlkn	|- ( N WSPathsN G ) C_ ( N WWalksN G )
3	2	a1i	|- ( ( Vtx ` G ) e. Fin -> ( N WSPathsN G ) C_ ( N WWalksN G ) )
4	1 3	ssfid	|- ( ( Vtx ` G ) e. Fin -> ( N WSPathsN G ) e. Fin )