Metamath Proof Explorer


Theorem spthonisspth

Description: A simple path between to vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018) (Revised by AV, 18-Jan-2021)

Ref Expression
Assertion spthonisspth
|- ( F ( A ( SPathsOn ` G ) B ) P -> F ( SPaths ` G ) P )

Proof

Step Hyp Ref Expression
1 eqid
 |-  ( Vtx ` G ) = ( Vtx ` G )
2 1 spthonprop
 |-  ( F ( A ( SPathsOn ` 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 ( SPaths ` 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 ( SPaths ` G ) P ) ) -> F ( SPaths ` G ) P )
4 2 3 syl
 |-  ( F ( A ( SPathsOn ` G ) B ) P -> F ( SPaths ` G ) P )