Metamath Proof Explorer

Theorem wlkoniswlk

Description: A walk between two vertices is a walk. (Contributed by Alexander van der Vekens, 12-Dec-2017) (Revised by AV, 2-Jan-2021)

Ref Expression
Assertion wlkoniswlk 
|- ( F ( A ( WalksOn ` G ) B ) P -> F ( Walks ` G ) P )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( Vtx ` G ) = ( Vtx ` G )
2 1 wlkonprop 
 |-  ( F ( A ( WalksOn ` G ) B ) P -> ( ( G e. _V /\ A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F ( Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) )
3 simp31 
 |-  ( ( ( G e. _V /\ A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F ( Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) -> F ( Walks ` G ) P )
4 2 3 syl 
 |-  ( F ( A ( WalksOn ` G ) B ) P -> F ( Walks ` G ) P )