Step |
Hyp |
Ref |
Expression |
1 |
|
2wlkd.p |
|- P = <" A B C "> |
2 |
|
2wlkd.f |
|- F = <" J K "> |
3 |
|
2wlkd.s |
|- ( ph -> ( A e. V /\ B e. V /\ C e. V ) ) |
4 |
|
2wlkd.n |
|- ( ph -> ( A =/= B /\ B =/= C ) ) |
5 |
|
2wlkd.e |
|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) ) |
6 |
|
2wlkd.v |
|- V = ( Vtx ` G ) |
7 |
|
2wlkd.i |
|- I = ( iEdg ` G ) |
8 |
|
2trld.n |
|- ( ph -> J =/= K ) |
9 |
1 2 3 4 5 6 7
|
2wlkd |
|- ( ph -> F ( Walks ` G ) P ) |
10 |
1 2 3 4 5
|
2wlkdlem7 |
|- ( ph -> ( J e. _V /\ K e. _V ) ) |
11 |
|
df-3an |
|- ( ( J e. _V /\ K e. _V /\ J =/= K ) <-> ( ( J e. _V /\ K e. _V ) /\ J =/= K ) ) |
12 |
10 8 11
|
sylanbrc |
|- ( ph -> ( J e. _V /\ K e. _V /\ J =/= K ) ) |
13 |
|
funcnvs2 |
|- ( ( J e. _V /\ K e. _V /\ J =/= K ) -> Fun `' <" J K "> ) |
14 |
12 13
|
syl |
|- ( ph -> Fun `' <" J K "> ) |
15 |
2
|
cnveqi |
|- `' F = `' <" J K "> |
16 |
15
|
funeqi |
|- ( Fun `' F <-> Fun `' <" J K "> ) |
17 |
14 16
|
sylibr |
|- ( ph -> Fun `' F ) |
18 |
|
istrl |
|- ( F ( Trails ` G ) P <-> ( F ( Walks ` G ) P /\ Fun `' F ) ) |
19 |
9 17 18
|
sylanbrc |
|- ( ph -> F ( Trails ` G ) P ) |