Theorem umgredgss

Description: The set of edges of a multigraph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 25-Nov-2020)

	Ref	Expression
Assertion	umgredgss	|- ( G e. UMGraph -> ( Edg ` G ) C_ { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } )

Proof

Step	Hyp	Ref	Expression
1		edgval	|- ( Edg ` G ) = ran ( iEdg ` G )
2		eqid	|- ( Vtx ` G ) = ( Vtx ` G )
3		eqid	|- ( iEdg ` G ) = ( iEdg ` G )
4	2 3	umgrf	|- ( G e. UMGraph -> ( iEdg ` G ) : dom ( iEdg ` G ) --> { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } )
5	4	frnd	|- ( G e. UMGraph -> ran ( iEdg ` G ) C_ { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } )
6	1 5	eqsstrid	|- ( G e. UMGraph -> ( Edg ` G ) C_ { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } )