Step |
Hyp |
Ref |
Expression |
1 |
|
simpr |
|- ( ( ( G e. W /\ ( Vtx ` G ) = (/) ) /\ k e. NN0* ) -> k e. NN0* ) |
2 |
|
rzal |
|- ( ( Vtx ` G ) = (/) -> A. v e. ( Vtx ` G ) ( ( VtxDeg ` G ) ` v ) = k ) |
3 |
2
|
ad2antlr |
|- ( ( ( G e. W /\ ( Vtx ` G ) = (/) ) /\ k e. NN0* ) -> A. v e. ( Vtx ` G ) ( ( VtxDeg ` G ) ` v ) = k ) |
4 |
|
eqid |
|- ( Vtx ` G ) = ( Vtx ` G ) |
5 |
|
eqid |
|- ( VtxDeg ` G ) = ( VtxDeg ` G ) |
6 |
4 5
|
isrgr |
|- ( ( G e. W /\ k e. NN0* ) -> ( G RegGraph k <-> ( k e. NN0* /\ A. v e. ( Vtx ` G ) ( ( VtxDeg ` G ) ` v ) = k ) ) ) |
7 |
6
|
adantlr |
|- ( ( ( G e. W /\ ( Vtx ` G ) = (/) ) /\ k e. NN0* ) -> ( G RegGraph k <-> ( k e. NN0* /\ A. v e. ( Vtx ` G ) ( ( VtxDeg ` G ) ` v ) = k ) ) ) |
8 |
1 3 7
|
mpbir2and |
|- ( ( ( G e. W /\ ( Vtx ` G ) = (/) ) /\ k e. NN0* ) -> G RegGraph k ) |
9 |
8
|
ralrimiva |
|- ( ( G e. W /\ ( Vtx ` G ) = (/) ) -> A. k e. NN0* G RegGraph k ) |