Step |
Hyp |
Ref |
Expression |
1 |
|
cyclispth |
|- ( F ( Cycles ` G ) P -> F ( Paths ` G ) P ) |
2 |
|
pthonpth |
|- ( F ( Paths ` G ) P -> F ( ( P ` 0 ) ( PathsOn ` G ) ( P ` ( # ` F ) ) ) P ) |
3 |
1 2
|
syl |
|- ( F ( Cycles ` G ) P -> F ( ( P ` 0 ) ( PathsOn ` G ) ( P ` ( # ` F ) ) ) P ) |
4 |
|
iscycl |
|- ( F ( Cycles ` G ) P <-> ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) ) |
5 |
4
|
simprbi |
|- ( F ( Cycles ` G ) P -> ( P ` 0 ) = ( P ` ( # ` F ) ) ) |
6 |
5
|
oveq2d |
|- ( F ( Cycles ` G ) P -> ( ( P ` 0 ) ( PathsOn ` G ) ( P ` 0 ) ) = ( ( P ` 0 ) ( PathsOn ` G ) ( P ` ( # ` F ) ) ) ) |
7 |
6
|
breqd |
|- ( F ( Cycles ` G ) P -> ( F ( ( P ` 0 ) ( PathsOn ` G ) ( P ` 0 ) ) P <-> F ( ( P ` 0 ) ( PathsOn ` G ) ( P ` ( # ` F ) ) ) P ) ) |
8 |
3 7
|
mpbird |
|- ( F ( Cycles ` G ) P -> F ( ( P ` 0 ) ( PathsOn ` G ) ( P ` 0 ) ) P ) |