Description: A path between two vertices is a trail between these vertices. (Contributed by AV, 24-Jan-2021)
Ref | Expression | ||
---|---|---|---|
Assertion | pthontrlon | |- ( F ( A ( PathsOn ` G ) B ) P -> F ( A ( TrailsOn ` G ) B ) P ) |
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 | simp3l | |- ( ( ( 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 ( A ( TrailsOn ` G ) B ) P ) |
|
4 | 2 3 | syl | |- ( F ( A ( PathsOn ` G ) B ) P -> F ( A ( TrailsOn ` G ) B ) P ) |