| 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 ) ) |