Theorem upgrfn

Description: The edge function of an undirected pseudograph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 11-Mar-2015) (Revised by AV, 10-Oct-2020)

	Ref	Expression
Hypotheses	isupgr.v	|- V = ( Vtx ` G )
	isupgr.e	|- E = ( iEdg ` G )
Assertion	upgrfn	|- ( ( G e. UPGraph /\ E Fn A ) -> E : A --> { 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	upgrf	|- ( G e. UPGraph -> E : dom E --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
4		fndm	|- ( E Fn A -> dom E = A )
5	4	feq2d	|- ( E Fn A -> ( E : dom E --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } <-> E : A --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } ) )
6	3 5	syl5ibcom	|- ( G e. UPGraph -> ( E Fn A -> E : A --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } ) )
7	6	imp	|- ( ( G e. UPGraph /\ E Fn A ) -> E : A --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )