Intersection of Image with Subset of Codomain

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

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

Let $A \subseteq S$ and $B \subseteq T$.

Then:
 * $f \left[{A \cap f^{-1} \left[{B}\right]}\right] = f \left[{A}\right] \cap B$