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Metamath Proof Explorer

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 ) e. Fin -> ( A ( N WWalksNOn G ) B ) e. Fin )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( Vtx ` G ) = ( Vtx ` G )
2 1 iswwlksnon 
 |-  ( A ( N WWalksNOn G ) B ) = { w e. ( N WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` N ) = B ) }
3 wwlksnfi 
 |-  ( ( Vtx ` G ) e. Fin -> ( N WWalksN G ) e. Fin )
4 rabfi 
 |-  ( ( N WWalksN G ) e. Fin -> { w e. ( N WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` N ) = B ) } e. Fin )
5 3 4 syl 
 |-  ( ( Vtx ` G ) e. Fin -> { w e. ( N WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` N ) = B ) } e. Fin )
6 2 5 eqeltrid 
 |-  ( ( Vtx ` G ) e. Fin -> ( A ( N WWalksNOn G ) B ) e. Fin )