Theorem umgrfn

Description: The edge function of an undirected multigraph is a function into unordered pairs of vertices. (Contributed by AV, 24-Nov-2020)

	Ref	Expression
Hypotheses	isumgr.v	|- V = ( Vtx ` G )
	isumgr.e	|- E = ( iEdg ` G )
Assertion	umgrfn	|- ( ( G e. UMGraph /\ E Fn A ) -> E : A --> { x e. ~P V | ( # ` x ) = 2 } )

Proof

Step	Hyp	Ref	Expression
1		isumgr.v	|- V = ( Vtx ` G )
2		isumgr.e	|- E = ( iEdg ` G )
3	1 2	umgrf	|- ( G e. UMGraph -> 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. UMGraph -> ( E Fn A -> E : A --> { x e. ~P V | ( # ` x ) = 2 } ) )
7	6	imp	|- ( ( G e. UMGraph /\ E Fn A ) -> E : A --> { x e. ~P V | ( # ` x ) = 2 } )