Intersection of Zero Loci is Zero Locus

Theorem
Let $k$ be a field.

Let $n \in \N_{>0}$.

Let $A = k \sqbrk {X_1, \ldots, X_n}$ be the ring of polynomials in $n$ variables over $k$.

Let $\mathbb S \subseteq \powerset A$ be a subset of the power set of $A$.

Then:
 * $\ds \bigcap _{S \mathop \in \mathbb S} \map V S = \map V {\bigcup \mathbb S}$

where $\map V \cdot$ denotes the zero locus.