Zero Locus of Vanishing Ideal is Closure

Theorem
Let $k$ be a field.

Let $n \ge 1$ be a natural number.

Let $\mathbb A^n_k$ be the standard affine space over $k$.

Let $\mathbb A^n_k$ be equipped by Zariski topology.

Let $S \subseteq \mathbb A^n_k$.

Then:
 * $\map V {\map I S} = S^-$

where:
 * $\map I \cdot$ denotes the vanishing ideal
 * $\map V \cdot$ denotes the zero locus
 * $S^-$ is the closure of $S$