Theorem uspgrupgrushgr

Description: A graph is a simple pseudograph iff it is a pseudograph and a simple hypergraph. (Contributed by AV, 30-Nov-2020)

	Ref	Expression
Assertion	uspgrupgrushgr	|- ( G e. USPGraph <-> ( G e. UPGraph /\ G e. USHGraph ) )

Proof

Step	Hyp	Ref	Expression
1		uspgrupgr	|- ( G e. USPGraph -> G e. UPGraph )
2		uspgrushgr	|- ( G e. USPGraph -> G e. USHGraph )
3	1 2	jca	|- ( G e. USPGraph -> ( G e. UPGraph /\ G e. USHGraph ) )
4		eqid	|- ( Vtx ` G ) = ( Vtx ` G )
5		eqid	|- ( iEdg ` G ) = ( iEdg ` G )
6	4 5	ushgrf	|- ( G e. USHGraph -> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> ( ~P ( Vtx ` G ) \ { (/) } ) )
7		edgval	|- ( Edg ` G ) = ran ( iEdg ` G )
8		upgredgss	|- ( G e. UPGraph -> ( Edg ` G ) C_ { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } )
9	7 8	eqsstrrid	|- ( G e. UPGraph -> ran ( iEdg ` G ) C_ { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } )
10		f1ssr	|- ( ( ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> ( ~P ( Vtx ` G ) \ { (/) } ) /\ ran ( iEdg ` G ) C_ { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } ) -> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } )
11	6 9 10	syl2anr	|- ( ( G e. UPGraph /\ G e. USHGraph ) -> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } )
12	4 5	isuspgr	|- ( G e. UPGraph -> ( G e. USPGraph <-> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } ) )
13	12	adantr	|- ( ( G e. UPGraph /\ G e. USHGraph ) -> ( G e. USPGraph <-> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } ) )
14	11 13	mpbird	|- ( ( G e. UPGraph /\ G e. USHGraph ) -> G e. USPGraph )
15	3 14	impbii	|- ( G e. USPGraph <-> ( G e. UPGraph /\ G e. USHGraph ) )