\begin{definition}[Definition:Direct Image Mapping/Relation]
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 $S \times T$.
The '''direct image mapping''' of $\RR$ is the mapping $\RR^\to: \powerset S \to \powerset T$ that sends a subset $X \subseteq T$ to its image under $\RR$:
:$\forall X \in \powerset S: \map {\RR^\to} X = \begin {cases} \set {t \in T: \exists s \in X: \tuple {s, t} \in \RR} & : X \ne \O \\ \O & : X = \O \end {cases}$
\end{definition}