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 ) ∈ Fin → ( 𝑁 WSPathsN 𝐺 ) ∈ Fin )

Proof

Step	Hyp	Ref	Expression
1		wwlksnfi	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( 𝑁 WWalksN 𝐺 ) ∈ Fin )
2		wspthsswwlkn	⊢ ( 𝑁 WSPathsN 𝐺 ) ⊆ ( 𝑁 WWalksN 𝐺 )
3	2	a1i	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( 𝑁 WSPathsN 𝐺 ) ⊆ ( 𝑁 WWalksN 𝐺 ) )
4	1 3	ssfid	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( 𝑁 WSPathsN 𝐺 ) ∈ Fin )