Theorem usgrumgruspgr

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

	Ref	Expression
Assertion	usgrumgruspgr	|- ( G e. USGraph <-> ( G e. UMGraph /\ G e. USPGraph ) )

Proof

Step	Hyp	Ref	Expression
1		usgrumgr	|- ( G e. USGraph -> G e. UMGraph )
2		usgruspgr	|- ( G e. USGraph -> G e. USPGraph )
3	1 2	jca	|- ( G e. USGraph -> ( G e. UMGraph /\ G e. USPGraph ) )
4		eqid	|- ( Vtx ` G ) = ( Vtx ` G )
5		eqid	|- ( iEdg ` G ) = ( iEdg ` G )
6	4 5	uspgrf	|- ( G e. USPGraph -> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } )
7		umgredgss	|- ( G e. UMGraph -> ( Edg ` G ) C_ { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } )
8		edgval	|- ( Edg ` G ) = ran ( iEdg ` G )
9		prprrab	|- { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } = { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 }
10	9	eqcomi	|- { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } = { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 }
11	7 8 10	3sstr3g	|- ( G e. UMGraph -> ran ( iEdg ` G ) C_ { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } )
12		f1ssr	|- ( ( ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } /\ 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 } )
13	6 11 12	syl2anr	|- ( ( G e. UMGraph /\ G e. USPGraph ) -> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } )
14	4 5	isusgr	|- ( G e. UMGraph -> ( G e. USGraph <-> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } ) )
15	14	adantr	|- ( ( G e. UMGraph /\ G e. USPGraph ) -> ( G e. USGraph <-> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } ) )
16	13 15	mpbird	|- ( ( G e. UMGraph /\ G e. USPGraph ) -> G e. USGraph )
17	3 16	impbii	|- ( G e. USGraph <-> ( G e. UMGraph /\ G e. USPGraph ) )