Theorem usgredgprv

Description: In a simple graph, an edge is an unordered pair of vertices. (Contributed by Alexander van der Vekens, 19-Aug-2017) (Revised by AV, 16-Oct-2020) (Proof shortened by AV, 11-Dec-2020)

	Ref	Expression
Hypotheses	usgredg2.e	|- E = ( iEdg ` G )
	usgredgprv.v	|- V = ( Vtx ` G )
Assertion	usgredgprv	|- ( ( G e. USGraph /\ X e. dom E ) -> ( ( E ` X ) = { M , N } -> ( M e. V /\ N e. V ) ) )

Proof

Step	Hyp	Ref	Expression
1		usgredg2.e	|- E = ( iEdg ` G )
2		usgredgprv.v	|- V = ( Vtx ` G )
3		usgrumgr	|- ( G e. USGraph -> G e. UMGraph )
4	1 2	umgredgprv	|- ( ( G e. UMGraph /\ X e. dom E ) -> ( ( E ` X ) = { M , N } -> ( M e. V /\ N e. V ) ) )
5	3 4	sylan	|- ( ( G e. USGraph /\ X e. dom E ) -> ( ( E ` X ) = { M , N } -> ( M e. V /\ N e. V ) ) )