| Step |
Hyp |
Ref |
Expression |
| 1 |
|
stgrvtx0.g |
|- G = ( StarGr ` N ) |
| 2 |
|
stgrvtx0.v |
|- V = ( Vtx ` G ) |
| 3 |
|
stgrvtx |
|- ( N e. NN0 -> ( Vtx ` ( StarGr ` N ) ) = ( 0 ... N ) ) |
| 4 |
1
|
fveq2i |
|- ( Vtx ` G ) = ( Vtx ` ( StarGr ` N ) ) |
| 5 |
2 4
|
eqtri |
|- V = ( Vtx ` ( StarGr ` N ) ) |
| 6 |
5
|
eqeq1i |
|- ( V = ( 0 ... N ) <-> ( Vtx ` ( StarGr ` N ) ) = ( 0 ... N ) ) |
| 7 |
|
0elfz |
|- ( N e. NN0 -> 0 e. ( 0 ... N ) ) |
| 8 |
|
eleq2 |
|- ( V = ( 0 ... N ) -> ( 0 e. V <-> 0 e. ( 0 ... N ) ) ) |
| 9 |
7 8
|
syl5ibrcom |
|- ( N e. NN0 -> ( V = ( 0 ... N ) -> 0 e. V ) ) |
| 10 |
6 9
|
biimtrrid |
|- ( N e. NN0 -> ( ( Vtx ` ( StarGr ` N ) ) = ( 0 ... N ) -> 0 e. V ) ) |
| 11 |
3 10
|
mpd |
|- ( N e. NN0 -> 0 e. V ) |