Direct Image Mapping is Mapping

Theorem

Let $S$ and $T$ be sets.

Let $\RR \subseteq S \times T$ be a relation on $S \times T$.

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

$\forall X \in \powerset S: \map {\RR^\to} X = \set {t \in T: \exists s \in X: \tuple {s, t} \in \RR}$

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

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.