Metamath Proof Explorer


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 } )