Theorem nbusgrvtx

Description: The set of neighbors of a vertex in a simple graph. (Contributed by Alexander van der Vekens, 9-Oct-2017) (Revised by AV, 26-Oct-2020) (Proof shortened by AV, 27-Nov-2020)

	Ref	Expression
Hypotheses	nbuhgr.v	|- V = ( Vtx ` G )
	nbuhgr.e	|- E = ( Edg ` G )
Assertion	nbusgrvtx	|- ( ( G e. USGraph /\ N e. V ) -> ( G NeighbVtx N ) = { n e. V | { N , n } e. E } )

Proof

Step	Hyp	Ref	Expression
1		nbuhgr.v	|- V = ( Vtx ` G )
2		nbuhgr.e	|- E = ( Edg ` G )
3		usgrumgr	|- ( G e. USGraph -> G e. UMGraph )
4	1 2	nbumgrvtx	|- ( ( G e. UMGraph /\ N e. V ) -> ( G NeighbVtx N ) = { n e. V | { N , n } e. E } )
5	3 4	sylan	|- ( ( G e. USGraph /\ N e. V ) -> ( G NeighbVtx N ) = { n e. V | { N , n } e. E } )