Step |
Hyp |
Ref |
Expression |
1 |
|
hashnbusgrvd.v |
|- V = ( Vtx ` G ) |
2 |
1
|
uvtxnbvtxm1 |
|- ( ( G e. FinUSGraph /\ U e. V ) -> ( U e. ( UnivVtx ` G ) <-> ( # ` ( G NeighbVtx U ) ) = ( ( # ` V ) - 1 ) ) ) |
3 |
|
fusgrusgr |
|- ( G e. FinUSGraph -> G e. USGraph ) |
4 |
1
|
hashnbusgrvd |
|- ( ( G e. USGraph /\ U e. V ) -> ( # ` ( G NeighbVtx U ) ) = ( ( VtxDeg ` G ) ` U ) ) |
5 |
3 4
|
sylan |
|- ( ( G e. FinUSGraph /\ U e. V ) -> ( # ` ( G NeighbVtx U ) ) = ( ( VtxDeg ` G ) ` U ) ) |
6 |
5
|
eqeq1d |
|- ( ( G e. FinUSGraph /\ U e. V ) -> ( ( # ` ( G NeighbVtx U ) ) = ( ( # ` V ) - 1 ) <-> ( ( VtxDeg ` G ) ` U ) = ( ( # ` V ) - 1 ) ) ) |
7 |
2 6
|
bitrd |
|- ( ( G e. FinUSGraph /\ U e. V ) -> ( U e. ( UnivVtx ` G ) <-> ( ( VtxDeg ` G ) ` U ) = ( ( # ` V ) - 1 ) ) ) |