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Metamath Proof Explorer

Theorem nbgrssvtx

Description: The neighbors of a vertex K in a graph form a subset of all vertices of the graph. (Contributed by Alexander van der Vekens, 12-Oct-2017) (Revised by AV, 26-Oct-2020) (Revised by AV, 12-Feb-2022)

		Ref	Expression
	Hypothesis	nbgrisvtx.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	Assertion	nbgrssvtx	⊢ ( 𝐺 NeighbVtx 𝐾 ) ⊆ 𝑉

Proof

Step	Hyp	Ref	Expression
1		nbgrisvtx.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2	1	nbgrisvtx	⊢ ( 𝑛 ∈ ( 𝐺 NeighbVtx 𝐾 ) → 𝑛 ∈ 𝑉 )
3	2	ssriv	⊢ ( 𝐺 NeighbVtx 𝐾 ) ⊆ 𝑉