Direct Image Mapping of Domain is Image Set of Relation

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

Let $\powerset S$ and $\powerset T$ be their power 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 $\RR$:


 * $\forall X \in \powerset S: \map {\RR^\to} X = \begin {cases} \set {t \in T: \exists s \in X: \tuple {x, t} \in \RR} & : X \ne \O \\ \O & : X = \O \end {cases}$

Then:
 * $\map {\RR^\to} {\Dom \RR} = \Img \RR$

where:
 * $\Dom \RR$ is the domain of $\RR$
 * $\Img \RR$ is the image set of $\RR$.