Step |
Hyp |
Ref |
Expression |
1 |
|
hashnbusgrvd.v |
|- V = ( Vtx ` G ) |
2 |
1
|
uvtxisvtx |
|- ( n e. ( UnivVtx ` G ) -> n e. V ) |
3 |
|
fveqeq2 |
|- ( v = n -> ( ( ( VtxDeg ` G ) ` v ) = ( ( # ` V ) - 1 ) <-> ( ( VtxDeg ` G ) ` n ) = ( ( # ` V ) - 1 ) ) ) |
4 |
3
|
rspccv |
|- ( A. v e. V ( ( VtxDeg ` G ) ` v ) = ( ( # ` V ) - 1 ) -> ( n e. V -> ( ( VtxDeg ` G ) ` n ) = ( ( # ` V ) - 1 ) ) ) |
5 |
4
|
adantl |
|- ( ( G e. FinUSGraph /\ A. v e. V ( ( VtxDeg ` G ) ` v ) = ( ( # ` V ) - 1 ) ) -> ( n e. V -> ( ( VtxDeg ` G ) ` n ) = ( ( # ` V ) - 1 ) ) ) |
6 |
5
|
imp |
|- ( ( ( G e. FinUSGraph /\ A. v e. V ( ( VtxDeg ` G ) ` v ) = ( ( # ` V ) - 1 ) ) /\ n e. V ) -> ( ( VtxDeg ` G ) ` n ) = ( ( # ` V ) - 1 ) ) |
7 |
1
|
usgruvtxvdb |
|- ( ( G e. FinUSGraph /\ n e. V ) -> ( n e. ( UnivVtx ` G ) <-> ( ( VtxDeg ` G ) ` n ) = ( ( # ` V ) - 1 ) ) ) |
8 |
7
|
adantlr |
|- ( ( ( G e. FinUSGraph /\ A. v e. V ( ( VtxDeg ` G ) ` v ) = ( ( # ` V ) - 1 ) ) /\ n e. V ) -> ( n e. ( UnivVtx ` G ) <-> ( ( VtxDeg ` G ) ` n ) = ( ( # ` V ) - 1 ) ) ) |
9 |
6 8
|
mpbird |
|- ( ( ( G e. FinUSGraph /\ A. v e. V ( ( VtxDeg ` G ) ` v ) = ( ( # ` V ) - 1 ) ) /\ n e. V ) -> n e. ( UnivVtx ` G ) ) |
10 |
9
|
ex |
|- ( ( G e. FinUSGraph /\ A. v e. V ( ( VtxDeg ` G ) ` v ) = ( ( # ` V ) - 1 ) ) -> ( n e. V -> n e. ( UnivVtx ` G ) ) ) |
11 |
2 10
|
impbid2 |
|- ( ( G e. FinUSGraph /\ A. v e. V ( ( VtxDeg ` G ) ` v ) = ( ( # ` V ) - 1 ) ) -> ( n e. ( UnivVtx ` G ) <-> n e. V ) ) |
12 |
11
|
eqrdv |
|- ( ( G e. FinUSGraph /\ A. v e. V ( ( VtxDeg ` G ) ` v ) = ( ( # ` V ) - 1 ) ) -> ( UnivVtx ` G ) = V ) |
13 |
|
fusgrusgr |
|- ( G e. FinUSGraph -> G e. USGraph ) |
14 |
1
|
cusgruvtxb |
|- ( G e. USGraph -> ( G e. ComplUSGraph <-> ( UnivVtx ` G ) = V ) ) |
15 |
13 14
|
syl |
|- ( G e. FinUSGraph -> ( G e. ComplUSGraph <-> ( UnivVtx ` G ) = V ) ) |
16 |
15
|
adantr |
|- ( ( G e. FinUSGraph /\ A. v e. V ( ( VtxDeg ` G ) ` v ) = ( ( # ` V ) - 1 ) ) -> ( G e. ComplUSGraph <-> ( UnivVtx ` G ) = V ) ) |
17 |
12 16
|
mpbird |
|- ( ( G e. FinUSGraph /\ A. v e. V ( ( VtxDeg ` G ) ` v ) = ( ( # ` V ) - 1 ) ) -> G e. ComplUSGraph ) |
18 |
17
|
ex |
|- ( G e. FinUSGraph -> ( A. v e. V ( ( VtxDeg ` G ) ` v ) = ( ( # ` V ) - 1 ) -> G e. ComplUSGraph ) ) |