Theorem umgrf

Description: The edge function of an undirected multigraph is a function into unordered pairs of vertices. Version of umgrfn without explicitly specified domain of the edge function. (Contributed by AV, 24-Nov-2020)

	Ref	Expression
Hypotheses	isumgr.v	|- V = ( Vtx ` G )
	isumgr.e	|- E = ( iEdg ` G )
Assertion	umgrf	|- ( G e. UMGraph -> E : dom E --> { x e. ~P V | ( # ` x ) = 2 } )

Proof

Step	Hyp	Ref	Expression
1		isumgr.v	|- V = ( Vtx ` G )
2		isumgr.e	|- E = ( iEdg ` G )
3	1 2	isumgrs	|- ( G e. UMGraph -> ( G e. UMGraph <-> E : dom E --> { x e. ~P V | ( # ` x ) = 2 } ) )
4	3	ibi	|- ( G e. UMGraph -> E : dom E --> { x e. ~P V | ( # ` x ) = 2 } )