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 ) ∈ Fin → ( 𝐴 ( 𝑁 WSPathsNOn 𝐺 ) 𝐵 ) ∈ Fin )

Proof

Step	Hyp	Ref	Expression
1		wwlksnonfi	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( 𝐴 ( 𝑁 WWalksNOn 𝐺 ) 𝐵 ) ∈ Fin )
2		wspthsswwlknon	⊢ ( 𝐴 ( 𝑁 WSPathsNOn 𝐺 ) 𝐵 ) ⊆ ( 𝐴 ( 𝑁 WWalksNOn 𝐺 ) 𝐵 )
3	2	a1i	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( 𝐴 ( 𝑁 WSPathsNOn 𝐺 ) 𝐵 ) ⊆ ( 𝐴 ( 𝑁 WWalksNOn 𝐺 ) 𝐵 ) )
4	1 3	ssfid	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( 𝐴 ( 𝑁 WSPathsNOn 𝐺 ) 𝐵 ) ∈ Fin )