Image of Intersection under Mapping/General Result

Theorem
Let $S$ and $T$ be sets.

Let $f: S \to T$ be a mapping.

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $\mathbb S \subseteq \mathcal P \left({S}\right)$.

Then:
 * $\displaystyle f \left[{\bigcap \mathbb S}\right] \subseteq \bigcap_{X \mathop \in \mathbb S} f \left[{X}\right]$

Proof
As $f$, being a mapping, is also a relation, we can apply Image of Intersection under Relation: General Result:


 * $\displaystyle \mathcal R \left[{\bigcap \mathbb S}\right] \subseteq \bigcap_{X \mathop \in \mathbb S} \mathcal R \left[{X}\right]$