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 $\mathcal R \subseteq S \times T$ be a relation on $S \times T$.

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


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

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

where:
 * $\Dom {\mathcal R}$ is the domain of $\mathcal R$
 * $\Img {\mathcal R}$ is the image set of $\mathcal R$.