Theorem usgredgss

Description: The set of edges of a simple graph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 1-Jan-2020) (Revised by AV, 14-Oct-2020)

	Ref	Expression
Assertion	usgredgss	|- ( G e. USGraph -> ( 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	usgrfs	|- ( G e. USGraph -> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } )
5		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 } )
6		frn	|- ( ( iEdg ` G ) : dom ( iEdg ` G ) --> { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } -> ran ( iEdg ` G ) C_ { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } )
7	4 5 6	3syl	|- ( G e. USGraph -> ran ( iEdg ` G ) C_ { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } )
8	1 7	eqsstrid	|- ( G e. USGraph -> ( Edg ` G ) C_ { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } )