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 )