Metamath Proof Explorer


Theorem upgruhgr

Description: An undirected pseudograph is an undirected hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017) (Revised by AV, 10-Oct-2020)

Ref Expression
Assertion upgruhgr
|- ( G e. UPGraph -> G e. UHGraph )

Proof

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 )