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 )