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Metamath Proof Explorer

Theorem trlsonfval

Description: The set of trails between two vertices. (Contributed by Alexander van der Vekens, 4-Nov-2017) (Revised by AV, 7-Jan-2021) (Proof shortened by AV, 15-Jan-2021) (Revised by AV, 21-Mar-2021)

Ref Expression
Hypothesis trlsonfval.v 
|- V = ( Vtx ` G )
Assertion trlsonfval 
|- ( ( A e. V /\ B e. V ) -> ( A ( TrailsOn ` G ) B ) = { <. f , p >. | ( f ( A ( WalksOn ` G ) B ) p /\ f ( Trails ` G ) p ) } )

Proof

Step Hyp Ref Expression
1 trlsonfval.v 
 |-  V = ( Vtx ` G )
2 1 1vgrex 
 |-  ( A e. V -> G e. _V )
3 2 adantr 
 |-  ( ( A e. V /\ B e. V ) -> G e. _V )
4 simpl 
 |-  ( ( A e. V /\ B e. V ) -> A e. V )
5 4 1 eleqtrdi 
 |-  ( ( A e. V /\ B e. V ) -> A e. ( Vtx ` G ) )
6 simpr 
 |-  ( ( A e. V /\ B e. V ) -> B e. V )
7 6 1 eleqtrdi 
 |-  ( ( A e. V /\ B e. V ) -> B e. ( Vtx ` G ) )
8 df-trlson 
 |-  TrailsOn = ( g e. _V |-> ( a e. ( Vtx ` g ) , b e. ( Vtx ` g ) |-> { <. f , p >. | ( f ( a ( WalksOn ` g ) b ) p /\ f ( Trails ` g ) p ) } ) )
9 3 5 7 8 mptmpoopabovd 
 |-  ( ( A e. V /\ B e. V ) -> ( A ( TrailsOn ` G ) B ) = { <. f , p >. | ( f ( A ( WalksOn ` G ) B ) p /\ f ( Trails ` G ) p ) } )