Theorem usgrf

Description: The edge function of a simple graph is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017) (Revised by AV, 13-Oct-2020)

	Ref	Expression
Hypotheses	isuspgr.v	|- V = ( Vtx ` G )
	isuspgr.e	|- E = ( iEdg ` G )
Assertion	usgrf	|- ( G e. USGraph -> E : dom E -1-1-> { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } )

Proof

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