Inverse Image Mapping of Codomain is Preimage 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 $\mathcal R^\gets: \powerset T \to \powerset S$ be the inverse 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:
 * $\map {\mathcal R^\gets} T = \Preimg {\mathcal R}$

where $\Preimg {\mathcal R}$ is the preimage of $\mathcal R$.