Theorem edgumgr

Description: Properties of an edge of a multigraph. (Contributed by AV, 25-Nov-2020)

	Ref	Expression
Assertion	edgumgr	|- ( ( G e. UMGraph /\ E e. ( Edg ` G ) ) -> ( E e. ~P ( Vtx ` G ) /\ ( # ` E ) = 2 ) )

Proof

Step	Hyp	Ref	Expression
1		umgredgss	|- ( G e. UMGraph -> ( Edg ` G ) C_ { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } )
2	1	sselda	|- ( ( G e. UMGraph /\ E e. ( Edg ` G ) ) -> E e. { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } )
3		fveqeq2	|- ( x = E -> ( ( # ` x ) = 2 <-> ( # ` E ) = 2 ) )
4	3	elrab	|- ( E e. { x e. ~P ( Vtx ` G ) | ( # ` x ) = 2 } <-> ( E e. ~P ( Vtx ` G ) /\ ( # ` E ) = 2 ) )
5	2 4	sylib	|- ( ( G e. UMGraph /\ E e. ( Edg ` G ) ) -> ( E e. ~P ( Vtx ` G ) /\ ( # ` E ) = 2 ) )