Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
|- ( Vtx ` G ) = ( Vtx ` G ) |
2 |
1
|
0pth |
|- ( G e. W -> ( (/) ( Paths ` G ) P <-> P : ( 0 ... 0 ) --> ( Vtx ` G ) ) ) |
3 |
2
|
anbi1d |
|- ( G e. W -> ( ( (/) ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` (/) ) ) ) <-> ( P : ( 0 ... 0 ) --> ( Vtx ` G ) /\ ( P ` 0 ) = ( P ` ( # ` (/) ) ) ) ) ) |
4 |
|
iscycl |
|- ( (/) ( Cycles ` G ) P <-> ( (/) ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` (/) ) ) ) ) |
5 |
|
hash0 |
|- ( # ` (/) ) = 0 |
6 |
5
|
eqcomi |
|- 0 = ( # ` (/) ) |
7 |
6
|
fveq2i |
|- ( P ` 0 ) = ( P ` ( # ` (/) ) ) |
8 |
7
|
biantru |
|- ( P : ( 0 ... 0 ) --> ( Vtx ` G ) <-> ( P : ( 0 ... 0 ) --> ( Vtx ` G ) /\ ( P ` 0 ) = ( P ` ( # ` (/) ) ) ) ) |
9 |
3 4 8
|
3bitr4g |
|- ( G e. W -> ( (/) ( Cycles ` G ) P <-> P : ( 0 ... 0 ) --> ( Vtx ` G ) ) ) |