| Step |
Hyp |
Ref |
Expression |
| 1 |
|
upgrres1.v |
|- V = ( Vtx ` G ) |
| 2 |
|
upgrres1.e |
|- E = ( Edg ` G ) |
| 3 |
|
upgrres1.f |
|- F = { e e. E | N e/ e } |
| 4 |
|
upgrres1.s |
|- S = <. ( V \ { N } ) , ( _I |` F ) >. |
| 5 |
4
|
fveq2i |
|- ( Vtx ` S ) = ( Vtx ` <. ( V \ { N } ) , ( _I |` F ) >. ) |
| 6 |
1 2 3
|
upgrres1lem1 |
|- ( ( V \ { N } ) e. _V /\ ( _I |` F ) e. _V ) |
| 7 |
|
opvtxfv |
|- ( ( ( V \ { N } ) e. _V /\ ( _I |` F ) e. _V ) -> ( Vtx ` <. ( V \ { N } ) , ( _I |` F ) >. ) = ( V \ { N } ) ) |
| 8 |
6 7
|
ax-mp |
|- ( Vtx ` <. ( V \ { N } ) , ( _I |` F ) >. ) = ( V \ { N } ) |
| 9 |
5 8
|
eqtri |
|- ( Vtx ` S ) = ( V \ { N } ) |