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	$$⊢V=\mathrm{Vtx}\left(G\right)$$
	nbuhgr.e	$$⊢E=\mathrm{Edg}\left(G\right)$$
Assertion	nbusgrvtx	$$⊢\left(G\in \mathrm{USGraph}\wedge N\in V\right)\to G\mathrm{NeighbVtx}N=\left\{n\in V|\left\{N,n\right\}\in E\right\}$$

Proof

Step	Hyp	Ref	Expression
1		nbuhgr.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
2		nbuhgr.e	$$⊢E=\mathrm{Edg}\left(G\right)$$
3		usgrumgr	$$⊢G\in \mathrm{USGraph}\to G\in \mathrm{UMGraph}$$
4	1 2	nbumgrvtx	$$⊢\left(G\in \mathrm{UMGraph}\wedge N\in V\right)\to G\mathrm{NeighbVtx}N=\left\{n\in V|\left\{N,n\right\}\in E\right\}$$
5	3 4	sylan	$$⊢\left(G\in \mathrm{USGraph}\wedge N\in V\right)\to G\mathrm{NeighbVtx}N=\left\{n\in V|\left\{N,n\right\}\in E\right\}$$