Step |
Hyp |
Ref |
Expression |
1 |
|
wksfval.v |
|- V = ( Vtx ` G ) |
2 |
|
wksfval.i |
|- I = ( iEdg ` G ) |
3 |
1 2
|
wksfval |
|- ( G e. _V -> ( Walks ` G ) = { <. f , p >. | ( f e. Word dom I /\ p : ( 0 ... ( # ` f ) ) --> V /\ A. k e. ( 0 ..^ ( # ` f ) ) if- ( ( p ` k ) = ( p ` ( k + 1 ) ) , ( I ` ( f ` k ) ) = { ( p ` k ) } , { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( I ` ( f ` k ) ) ) ) } ) |
4 |
3
|
brfvopab |
|- ( F ( Walks ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) ) |
5 |
1 2
|
iswlk |
|- ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( F ( Walks ` G ) P <-> ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0 ..^ ( # ` F ) ) if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) ) ) |
6 |
5
|
biimpd |
|- ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( F ( Walks ` G ) P -> ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0 ..^ ( # ` F ) ) if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) ) ) |
7 |
4 6
|
mpcom |
|- ( F ( Walks ` G ) P -> ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0 ..^ ( # ` F ) ) if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) ) |