Step |
Hyp |
Ref |
Expression |
1 |
|
id |
|- ( F ( Walks ` G ) P -> F ( Walks ` G ) P ) |
2 |
|
eqidd |
|- ( F ( Walks ` G ) P -> ( P ` 0 ) = ( P ` 0 ) ) |
3 |
|
eqidd |
|- ( F ( Walks ` G ) P -> ( P ` ( # ` F ) ) = ( P ` ( # ` F ) ) ) |
4 |
|
eqid |
|- ( Vtx ` G ) = ( Vtx ` G ) |
5 |
4
|
wlkepvtx |
|- ( F ( Walks ` G ) P -> ( ( P ` 0 ) e. ( Vtx ` G ) /\ ( P ` ( # ` F ) ) e. ( Vtx ` G ) ) ) |
6 |
|
wlkv |
|- ( F ( Walks ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) ) |
7 |
|
3simpc |
|- ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( F e. _V /\ P e. _V ) ) |
8 |
6 7
|
syl |
|- ( F ( Walks ` G ) P -> ( F e. _V /\ P e. _V ) ) |
9 |
4
|
iswlkon |
|- ( ( ( ( P ` 0 ) e. ( Vtx ` G ) /\ ( P ` ( # ` F ) ) e. ( Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( ( P ` 0 ) ( WalksOn ` G ) ( P ` ( # ` F ) ) ) P <-> ( F ( Walks ` G ) P /\ ( P ` 0 ) = ( P ` 0 ) /\ ( P ` ( # ` F ) ) = ( P ` ( # ` F ) ) ) ) ) |
10 |
5 8 9
|
syl2anc |
|- ( F ( Walks ` G ) P -> ( F ( ( P ` 0 ) ( WalksOn ` G ) ( P ` ( # ` F ) ) ) P <-> ( F ( Walks ` G ) P /\ ( P ` 0 ) = ( P ` 0 ) /\ ( P ` ( # ` F ) ) = ( P ` ( # ` F ) ) ) ) ) |
11 |
1 2 3 10
|
mpbir3and |
|- ( F ( Walks ` G ) P -> F ( ( P ` 0 ) ( WalksOn ` G ) ( P ` ( # ` F ) ) ) P ) |