Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
|- ( Vtx ` G ) = ( Vtx ` G ) |
2 |
|
eqid |
|- ( iEdg ` G ) = ( iEdg ` G ) |
3 |
1 2
|
upgrf |
|- ( G e. UPGraph -> ( iEdg ` G ) : dom ( iEdg ` G ) --> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } ) |
4 |
|
ssrab2 |
|- { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } C_ ( ~P ( Vtx ` G ) \ { (/) } ) |
5 |
|
fss |
|- ( ( ( iEdg ` G ) : dom ( iEdg ` G ) --> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } /\ { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } C_ ( ~P ( Vtx ` G ) \ { (/) } ) ) -> ( iEdg ` G ) : dom ( iEdg ` G ) --> ( ~P ( Vtx ` G ) \ { (/) } ) ) |
6 |
3 4 5
|
sylancl |
|- ( G e. UPGraph -> ( iEdg ` G ) : dom ( iEdg ` G ) --> ( ~P ( Vtx ` G ) \ { (/) } ) ) |
7 |
1 2
|
isuhgr |
|- ( G e. UPGraph -> ( G e. UHGraph <-> ( iEdg ` G ) : dom ( iEdg ` G ) --> ( ~P ( Vtx ` G ) \ { (/) } ) ) ) |
8 |
6 7
|
mpbird |
|- ( G e. UPGraph -> G e. UHGraph ) |