Theorem upgrf

Description: The edge function of an undirected pseudograph is a function into unordered pairs of vertices. Version of upgrfn without explicitly specified domain of the edge function. (Contributed by Mario Carneiro, 12-Mar-2015) (Revised by AV, 10-Oct-2020)

	Ref	Expression
Hypotheses	isupgr.v	|- V = ( Vtx ` G )
	isupgr.e	|- E = ( iEdg ` G )
Assertion	upgrf	|- ( G e. UPGraph -> E : dom E --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )

Proof

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