Step |
Hyp |
Ref |
Expression |
1 |
|
isspth |
|- ( F ( SPaths ` G ) P <-> ( F ( Trails ` G ) P /\ Fun `' P ) ) |
2 |
|
trliswlk |
|- ( F ( Trails ` G ) P -> F ( Walks ` G ) P ) |
3 |
|
eqid |
|- ( Vtx ` G ) = ( Vtx ` G ) |
4 |
3
|
wlkp |
|- ( F ( Walks ` G ) P -> P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) |
5 |
|
df-f1 |
|- ( P : ( 0 ... ( # ` F ) ) -1-1-> ( Vtx ` G ) <-> ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' P ) ) |
6 |
5
|
simplbi2 |
|- ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) -> ( Fun `' P -> P : ( 0 ... ( # ` F ) ) -1-1-> ( Vtx ` G ) ) ) |
7 |
2 4 6
|
3syl |
|- ( F ( Trails ` G ) P -> ( Fun `' P -> P : ( 0 ... ( # ` F ) ) -1-1-> ( Vtx ` G ) ) ) |
8 |
7
|
imp |
|- ( ( F ( Trails ` G ) P /\ Fun `' P ) -> P : ( 0 ... ( # ` F ) ) -1-1-> ( Vtx ` G ) ) |
9 |
1 8
|
sylbi |
|- ( F ( SPaths ` G ) P -> P : ( 0 ... ( # ` F ) ) -1-1-> ( Vtx ` G ) ) |