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

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
2	1	iswwlksnon	⊢ ( 𝐴 ( 𝑁 WWalksNOn 𝐺 ) 𝐵 ) = { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( ( 𝑤 ‘ 0 ) = 𝐴 ∧ ( 𝑤 ‘ 𝑁 ) = 𝐵 ) }
3		wwlksnfi	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( 𝑁 WWalksN 𝐺 ) ∈ Fin )
4		rabfi	⊢ ( ( 𝑁 WWalksN 𝐺 ) ∈ Fin → { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( ( 𝑤 ‘ 0 ) = 𝐴 ∧ ( 𝑤 ‘ 𝑁 ) = 𝐵 ) } ∈ Fin )
5	3 4	syl	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( ( 𝑤 ‘ 0 ) = 𝐴 ∧ ( 𝑤 ‘ 𝑁 ) = 𝐵 ) } ∈ Fin )
6	2 5	eqeltrid	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( 𝐴 ( 𝑁 WWalksNOn 𝐺 ) 𝐵 ) ∈ Fin )