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 } ) |
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 } ) |