Metamath Proof Explorer

Theorem pthonispth

Description: A path between two vertices is a path. (Contributed by Alexander van der Vekens, 12-Dec-2017) (Revised by AV, 17-Jan-2021)

Ref Expression
Assertion pthonispth 
|- ( F ( A ( PathsOn ` G ) B ) P -> F ( Paths ` G ) P )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( Vtx ` G ) = ( Vtx ` G )
2 1 pthsonprop 
 |-  ( F ( A ( PathsOn ` G ) B ) P -> ( ( G e. _V /\ A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F ( A ( TrailsOn ` G ) B ) P /\ F ( Paths ` G ) P ) ) )
3 simp3r 
 |-  ( ( ( G e. _V /\ A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F ( A ( TrailsOn ` G ) B ) P /\ F ( Paths ` G ) P ) ) -> F ( Paths ` G ) P )
4 2 3 syl 
 |-  ( F ( A ( PathsOn ` G ) B ) P -> F ( Paths ` G ) P )