Image of Intersection under Injection/General Result

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

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

Then:
 * $\displaystyle \forall \mathbb S \subseteq \mathcal P \left({S}\right): f \left({\bigcap \mathbb S}\right) = \bigcap_{X \mathop \in \mathbb S} f \left({X}\right)$

iff $f$ is an injection.

Proof
Follows directly from the same approach as Injection Image of Intersection, and from One-to-Many Image of Intersections: General Result.