Step |
Hyp |
Ref |
Expression |
1 |
|
nbupgruvtxres.v |
|- V = ( Vtx ` G ) |
2 |
|
nbupgruvtxres.e |
|- E = ( Edg ` G ) |
3 |
|
nbupgruvtxres.f |
|- F = { e e. E | N e/ e } |
4 |
|
nbupgruvtxres.s |
|- S = <. ( V \ { N } ) , ( _I |` F ) >. |
5 |
1
|
uvtxnbgr |
|- ( K e. ( UnivVtx ` G ) -> ( G NeighbVtx K ) = ( V \ { K } ) ) |
6 |
1 2 3 4
|
nbupgruvtxres |
|- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) -> ( ( G NeighbVtx K ) = ( V \ { K } ) -> ( S NeighbVtx K ) = ( V \ { N , K } ) ) ) |
7 |
6
|
imp |
|- ( ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) /\ ( G NeighbVtx K ) = ( V \ { K } ) ) -> ( S NeighbVtx K ) = ( V \ { N , K } ) ) |
8 |
|
difpr |
|- ( V \ { N , K } ) = ( ( V \ { N } ) \ { K } ) |
9 |
1 2 3 4
|
upgrres1lem2 |
|- ( Vtx ` S ) = ( V \ { N } ) |
10 |
9
|
difeq1i |
|- ( ( Vtx ` S ) \ { K } ) = ( ( V \ { N } ) \ { K } ) |
11 |
10
|
a1i |
|- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) -> ( ( Vtx ` S ) \ { K } ) = ( ( V \ { N } ) \ { K } ) ) |
12 |
8 11
|
eqtr4id |
|- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) -> ( V \ { N , K } ) = ( ( Vtx ` S ) \ { K } ) ) |
13 |
12
|
adantr |
|- ( ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) /\ ( G NeighbVtx K ) = ( V \ { K } ) ) -> ( V \ { N , K } ) = ( ( Vtx ` S ) \ { K } ) ) |
14 |
7 13
|
eqtrd |
|- ( ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) /\ ( G NeighbVtx K ) = ( V \ { K } ) ) -> ( S NeighbVtx K ) = ( ( Vtx ` S ) \ { K } ) ) |
15 |
|
simpr |
|- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) -> K e. ( V \ { N } ) ) |
16 |
15 9
|
eleqtrrdi |
|- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) -> K e. ( Vtx ` S ) ) |
17 |
16
|
adantr |
|- ( ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) /\ ( G NeighbVtx K ) = ( V \ { K } ) ) -> K e. ( Vtx ` S ) ) |
18 |
|
eqid |
|- ( Vtx ` S ) = ( Vtx ` S ) |
19 |
18
|
uvtxnbgrb |
|- ( K e. ( Vtx ` S ) -> ( K e. ( UnivVtx ` S ) <-> ( S NeighbVtx K ) = ( ( Vtx ` S ) \ { K } ) ) ) |
20 |
17 19
|
syl |
|- ( ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) /\ ( G NeighbVtx K ) = ( V \ { K } ) ) -> ( K e. ( UnivVtx ` S ) <-> ( S NeighbVtx K ) = ( ( Vtx ` S ) \ { K } ) ) ) |
21 |
14 20
|
mpbird |
|- ( ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) /\ ( G NeighbVtx K ) = ( V \ { K } ) ) -> K e. ( UnivVtx ` S ) ) |
22 |
21
|
ex |
|- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) -> ( ( G NeighbVtx K ) = ( V \ { K } ) -> K e. ( UnivVtx ` S ) ) ) |
23 |
5 22
|
syl5 |
|- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) -> ( K e. ( UnivVtx ` G ) -> K e. ( UnivVtx ` S ) ) ) |