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Metamath Proof Explorer

Theorem spthonpthon

Description: A simple path between two vertices is a path between these vertices. (Contributed by AV, 24-Jan-2021)

Ref Expression
Assertion spthonpthon 
|- ( F ( A ( SPathsOn ` G ) B ) P -> F ( A ( PathsOn ` G ) B ) 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 3simpc 
 |-  ( ( G e. _V /\ A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) -> ( A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) )
4 3 3anim1i 
 |-  ( ( ( 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 ) ) -> ( ( A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F ( A ( TrailsOn ` G ) B ) P /\ F ( SPaths ` G ) P ) ) )
5 spthispth 
 |-  ( F ( SPaths ` G ) P -> F ( Paths ` G ) P )
6 5 anim2i 
 |-  ( ( F ( A ( TrailsOn ` G ) B ) P /\ F ( SPaths ` G ) P ) -> ( F ( A ( TrailsOn ` G ) B ) P /\ F ( Paths ` G ) P ) )
7 6 3ad2ant3 
 |-  ( ( ( 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 ( A ( TrailsOn ` G ) B ) P /\ F ( Paths ` G ) P ) )
8 1 ispthson 
 |-  ( ( ( A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( A ( PathsOn ` G ) B ) P <-> ( F ( A ( TrailsOn ` G ) B ) P /\ F ( Paths ` G ) P ) ) )
9 8 3adant3 
 |-  ( ( ( 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 ( A ( PathsOn ` G ) B ) P <-> ( F ( A ( TrailsOn ` G ) B ) P /\ F ( Paths ` G ) P ) ) )
10 7 9 mpbird 
 |-  ( ( ( 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 ( A ( PathsOn ` G ) B ) P )
11 2 4 10 3syl 
 |-  ( F ( A ( SPathsOn ` G ) B ) P -> F ( A ( PathsOn ` G ) B ) P )