Theorem umgr0e

Description: The empty graph, with vertices but no edges, is a multigraph. (Contributed by Mario Carneiro, 12-Mar-2015) (Revised by AV, 25-Nov-2020)

	Ref	Expression
Hypotheses	umgr0e.g	|- ( ph -> G e. W )
	umgr0e.e	|- ( ph -> ( iEdg ` G ) = (/) )
Assertion	umgr0e	|- ( ph -> G e. UMGraph )

Proof

Step	Hyp	Ref	Expression
1		umgr0e.g	|- ( ph -> G e. W )
2		umgr0e.e	|- ( ph -> ( iEdg ` G ) = (/) )
3	2	f10d	|- ( ph -> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } )
4		f1f	|- ( ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } -> ( iEdg ` G ) : dom ( iEdg ` G ) --> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } )
5	3 4	syl	|- ( ph -> ( iEdg ` G ) : dom ( iEdg ` G ) --> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } )
6		eqid	|- ( Vtx ` G ) = ( Vtx ` G )
7		eqid	|- ( iEdg ` G ) = ( iEdg ` G )
8	6 7	isumgr	|- ( G e. W -> ( G e. UMGraph <-> ( iEdg ` G ) : dom ( iEdg ` G ) --> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } ) )
9	1 8	syl	|- ( ph -> ( G e. UMGraph <-> ( iEdg ` G ) : dom ( iEdg ` G ) --> { x e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` x ) = 2 } ) )
10	5 9	mpbird	|- ( ph -> G e. UMGraph )