Step |
Hyp |
Ref |
Expression |
1 |
|
trlsegvdeg.v |
|- V = ( Vtx ` G ) |
2 |
|
trlsegvdeg.i |
|- I = ( iEdg ` G ) |
3 |
|
trlsegvdeg.f |
|- ( ph -> Fun I ) |
4 |
|
trlsegvdeg.n |
|- ( ph -> N e. ( 0 ..^ ( # ` F ) ) ) |
5 |
|
trlsegvdeg.u |
|- ( ph -> U e. V ) |
6 |
|
trlsegvdeg.w |
|- ( ph -> F ( Trails ` G ) P ) |
7 |
|
trlsegvdeg.vx |
|- ( ph -> ( Vtx ` X ) = V ) |
8 |
|
trlsegvdeg.vy |
|- ( ph -> ( Vtx ` Y ) = V ) |
9 |
|
trlsegvdeg.vz |
|- ( ph -> ( Vtx ` Z ) = V ) |
10 |
|
trlsegvdeg.ix |
|- ( ph -> ( iEdg ` X ) = ( I |` ( F " ( 0 ..^ N ) ) ) ) |
11 |
|
trlsegvdeg.iy |
|- ( ph -> ( iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) |
12 |
|
trlsegvdeg.iz |
|- ( ph -> ( iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) ) |
13 |
|
fvex |
|- ( F ` N ) e. _V |
14 |
|
fvex |
|- ( I ` ( F ` N ) ) e. _V |
15 |
13 14
|
pm3.2i |
|- ( ( F ` N ) e. _V /\ ( I ` ( F ` N ) ) e. _V ) |
16 |
|
funsng |
|- ( ( ( F ` N ) e. _V /\ ( I ` ( F ` N ) ) e. _V ) -> Fun { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) |
17 |
15 16
|
mp1i |
|- ( ph -> Fun { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) |
18 |
11
|
funeqd |
|- ( ph -> ( Fun ( iEdg ` Y ) <-> Fun { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) ) |
19 |
17 18
|
mpbird |
|- ( ph -> Fun ( iEdg ` Y ) ) |