Image of Subset under Mapping equals Union of Images of Elements

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

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

Let $X \subseteq S$ be a subset of $S$.

Then:
 * $\displaystyle f \left[{X}\right] = \bigcup_{x \mathop \in X} f \left({x}\right)$

where:
 * $f \left[{X}\right]$ is the image of the subset $X$ under $f$
 * $f \left({x}\right)$ is the image of the element $x$ under $f$.

Proof
By definition, a mapping is a relation.

Thus Image of Subset under Relation equals Union of Images of Elements applies.