Direct Image Mapping of Mapping is Mapping

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

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

Let $f^\to: \powerset S \to \powerset T$ be the direct image mapping of $f$:


 * $\forall X \in \powerset S: \map {f^\to} X = \set {t \in T: \exists s \in X: \map f s = t}$

Then $f^\to$ is indeed a mapping.

Proof
$f$, being a mapping, is also a relation.

Hence Direct Image Mapping of Relation is Mapping can be applied directly.