Database
GRAPH THEORY
Undirected graphs
Neighbors, complete graphs and universal vertices
Neighbors
nbusgr
Metamath Proof Explorer
Description: The set of neighbors of an arbitrary class 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
nbusgr
⊢ ( 𝐺 ∈ USGraph → ( 𝐺 NeighbVtx 𝑁 ) = { 𝑛 ∈ 𝑉 ∣ { 𝑁 , 𝑛 } ∈ 𝐸 } )
Proof
Step
Hyp
Ref
Expression
1
nbuhgr.v
⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2
nbuhgr.e
⊢ 𝐸 = ( Edg ‘ 𝐺 )
3
usgrumgr
⊢ ( 𝐺 ∈ USGraph → 𝐺 ∈ UMGraph )
4
1 2
nbumgr
⊢ ( 𝐺 ∈ UMGraph → ( 𝐺 NeighbVtx 𝑁 ) = { 𝑛 ∈ 𝑉 ∣ { 𝑁 , 𝑛 } ∈ 𝐸 } )
5
3 4
syl
⊢ ( 𝐺 ∈ USGraph → ( 𝐺 NeighbVtx 𝑁 ) = { 𝑛 ∈ 𝑉 ∣ { 𝑁 , 𝑛 } ∈ 𝐸 } )