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Metamath Proof Explorer

Theorem usgredg2

Description: The value of the "edge function" of a simple graph is a set containing two elements (the vertices the corresponding edge is connecting). (Contributed by Alexander van der Vekens, 11-Aug-2017) (Revised by AV, 16-Oct-2020) (Proof shortened by AV, 11-Dec-2020)

		Ref	Expression
	Hypothesis	usgredg2.e	\|- E = ( iEdg ` G )
	Assertion	usgredg2	\|- ( ( G e. USGraph /\ X e. dom E ) -> ( # ` ( E ` X ) ) = 2 )

Proof

Step	Hyp	Ref	Expression
1		usgredg2.e	\|- E = ( iEdg ` G )
2		usgrumgr	\|- ( G e. USGraph -> G e. UMGraph )
3		eqid	\|- ( Vtx ` G ) = ( Vtx ` G )
4	3 1	umgredg2	\|- ( ( G e. UMGraph /\ X e. dom E ) -> ( # ` ( E ` X ) ) = 2 )
5	2 4	sylan	\|- ( ( G e. USGraph /\ X e. dom E ) -> ( # ` ( E ` X ) ) = 2 )