Intersection of Set whose Every Element is Closed under Mapping is also Closed under Mapping

Theorem
Let $S$ be a set of sets.

Let $g$ be a mapping such that:
 * for every $x \in S$, $x$ is closed under $g$.

Then the intersection $\ds \bigcap S$ of $S$ is also closed under $g$.

Also see

 * Intersection of Set whose Every Element is Closed under Chain Unions is also Closed under Chain Unions