Image of Intersection under Mapping/General Result

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

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

Let $\powerset S$ be the power set of $S$.

Let $\mathbb S \subseteq \powerset S$.

Then:
 * $\displaystyle f \sqbrk {\bigcap \mathbb S} \subseteq \bigcap_{X \mathop \in \mathbb S} f \sqbrk X$

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


 * $\displaystyle \RR \sqbrk {\bigcap \mathbb S} \subseteq \bigcap_{X \mathop \in \mathbb S} \RR \sqbrk X$