Theorem usgrf1o

Description: The edge function of a simple graph is a bijective function onto its range. (Contributed by Alexander van der Vekens, 18-Nov-2017) (Revised by AV, 15-Oct-2020)

	Ref	Expression
Hypothesis	usgrf1o.e	|- E = ( iEdg ` G )
Assertion	usgrf1o	|- ( G e. USGraph -> E : dom E -1-1-onto-> ran E )

Proof

Step	Hyp	Ref	Expression
1		usgrf1o.e	|- E = ( iEdg ` G )
2		eqid	|- ( Vtx ` G ) = ( Vtx ` G )
3	2 1	usgrfs	|- ( G e. USGraph -> E : dom E -1-1-> { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } )
4		f1f1orn	|- ( E : dom E -1-1-> { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } -> E : dom E -1-1-onto-> ran E )
5	3 4	syl	|- ( G e. USGraph -> E : dom E -1-1-onto-> ran E )