Metamath Proof Explorer

Theorem spthonprop

Description: Properties of a simple path between two vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018) (Revised by AV, 16-Jan-2021)

Ref Expression
Hypothesis pthsonfval.v 
|- V = ( Vtx ` G )
Assertion spthonprop 
|- ( F ( A ( SPathsOn ` G ) B ) P -> ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) /\ ( F ( A ( TrailsOn ` G ) B ) P /\ F ( SPaths ` G ) P ) ) )

Proof

Step Hyp Ref Expression
1 pthsonfval.v 
 |-  V = ( Vtx ` G )
2 1 isspthson 
 |-  ( ( ( A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( A ( SPathsOn ` G ) B ) P <-> ( F ( A ( TrailsOn ` G ) B ) P /\ F ( SPaths ` G ) P ) ) )
3 2 3adantl1 
 |-  ( ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( A ( SPathsOn ` G ) B ) P <-> ( F ( A ( TrailsOn ` G ) B ) P /\ F ( SPaths ` G ) P ) ) )
4 df-spthson 
 |-  SPathsOn = ( g e. _V |-> ( a e. ( Vtx ` g ) , b e. ( Vtx ` g ) |-> { <. f , p >. | ( f ( a ( TrailsOn ` g ) b ) p /\ f ( SPaths ` g ) p ) } ) )
5 spthiswlk 
 |-  ( f ( SPaths ` G ) p -> f ( Walks ` G ) p )
6 5 adantl 
 |-  ( ( ( G e. _V /\ A e. V /\ B e. V ) /\ f ( SPaths ` G ) p ) -> f ( Walks ` G ) p )
7 1 3 4 6 wksonproplem 
 |-  ( F ( A ( SPathsOn ` G ) B ) P -> ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) /\ ( F ( A ( TrailsOn ` G ) B ) P /\ F ( SPaths ` G ) P ) ) )