Step |
Hyp |
Ref |
Expression |
1 |
|
trliswlk |
|- ( F ( Trails ` G ) P -> F ( Walks ` G ) P ) |
2 |
|
wlkonwlk |
|- ( F ( Walks ` G ) P -> F ( ( P ` 0 ) ( WalksOn ` G ) ( P ` ( # ` F ) ) ) P ) |
3 |
1 2
|
syl |
|- ( F ( Trails ` G ) P -> F ( ( P ` 0 ) ( WalksOn ` G ) ( P ` ( # ` F ) ) ) P ) |
4 |
|
id |
|- ( F ( Trails ` G ) P -> F ( Trails ` G ) P ) |
5 |
|
eqid |
|- ( Vtx ` G ) = ( Vtx ` G ) |
6 |
5
|
wlkepvtx |
|- ( F ( Walks ` G ) P -> ( ( P ` 0 ) e. ( Vtx ` G ) /\ ( P ` ( # ` F ) ) e. ( Vtx ` G ) ) ) |
7 |
|
wlkv |
|- ( F ( Walks ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) ) |
8 |
|
3simpc |
|- ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( F e. _V /\ P e. _V ) ) |
9 |
8
|
anim2i |
|- ( ( ( ( P ` 0 ) e. ( Vtx ` G ) /\ ( P ` ( # ` F ) ) e. ( Vtx ` G ) ) /\ ( G e. _V /\ F e. _V /\ P e. _V ) ) -> ( ( ( P ` 0 ) e. ( Vtx ` G ) /\ ( P ` ( # ` F ) ) e. ( Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) ) ) |
10 |
6 7 9
|
syl2anc |
|- ( F ( Walks ` G ) P -> ( ( ( P ` 0 ) e. ( Vtx ` G ) /\ ( P ` ( # ` F ) ) e. ( Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) ) ) |
11 |
1 10
|
syl |
|- ( F ( Trails ` G ) P -> ( ( ( P ` 0 ) e. ( Vtx ` G ) /\ ( P ` ( # ` F ) ) e. ( Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) ) ) |
12 |
5
|
istrlson |
|- ( ( ( ( P ` 0 ) e. ( Vtx ` G ) /\ ( P ` ( # ` F ) ) e. ( Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( ( P ` 0 ) ( TrailsOn ` G ) ( P ` ( # ` F ) ) ) P <-> ( F ( ( P ` 0 ) ( WalksOn ` G ) ( P ` ( # ` F ) ) ) P /\ F ( Trails ` G ) P ) ) ) |
13 |
11 12
|
syl |
|- ( F ( Trails ` G ) P -> ( F ( ( P ` 0 ) ( TrailsOn ` G ) ( P ` ( # ` F ) ) ) P <-> ( F ( ( P ` 0 ) ( WalksOn ` G ) ( P ` ( # ` F ) ) ) P /\ F ( Trails ` G ) P ) ) ) |
14 |
3 4 13
|
mpbir2and |
|- ( F ( Trails ` G ) P -> F ( ( P ` 0 ) ( TrailsOn ` G ) ( P ` ( # ` F ) ) ) P ) |