Step |
Hyp |
Ref |
Expression |
1 |
|
trlsonfval.v |
|- V = ( Vtx ` G ) |
2 |
1
|
trlsonfval |
|- ( ( A e. V /\ B e. V ) -> ( A ( TrailsOn ` G ) B ) = { <. f , p >. | ( f ( A ( WalksOn ` G ) B ) p /\ f ( Trails ` G ) p ) } ) |
3 |
2
|
breqd |
|- ( ( A e. V /\ B e. V ) -> ( F ( A ( TrailsOn ` G ) B ) P <-> F { <. f , p >. | ( f ( A ( WalksOn ` G ) B ) p /\ f ( Trails ` G ) p ) } P ) ) |
4 |
|
breq12 |
|- ( ( f = F /\ p = P ) -> ( f ( A ( WalksOn ` G ) B ) p <-> F ( A ( WalksOn ` G ) B ) P ) ) |
5 |
|
breq12 |
|- ( ( f = F /\ p = P ) -> ( f ( Trails ` G ) p <-> F ( Trails ` G ) P ) ) |
6 |
4 5
|
anbi12d |
|- ( ( f = F /\ p = P ) -> ( ( f ( A ( WalksOn ` G ) B ) p /\ f ( Trails ` G ) p ) <-> ( F ( A ( WalksOn ` G ) B ) P /\ F ( Trails ` G ) P ) ) ) |
7 |
|
eqid |
|- { <. f , p >. | ( f ( A ( WalksOn ` G ) B ) p /\ f ( Trails ` G ) p ) } = { <. f , p >. | ( f ( A ( WalksOn ` G ) B ) p /\ f ( Trails ` G ) p ) } |
8 |
6 7
|
brabga |
|- ( ( F e. U /\ P e. Z ) -> ( F { <. f , p >. | ( f ( A ( WalksOn ` G ) B ) p /\ f ( Trails ` G ) p ) } P <-> ( F ( A ( WalksOn ` G ) B ) P /\ F ( Trails ` G ) P ) ) ) |
9 |
3 8
|
sylan9bb |
|- ( ( ( A e. V /\ B e. V ) /\ ( F e. U /\ P e. Z ) ) -> ( F ( A ( TrailsOn ` G ) B ) P <-> ( F ( A ( WalksOn ` G ) B ) P /\ F ( Trails ` G ) P ) ) ) |