Inverse Image Mapping of Codomain is Preimage Set of Mapping

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

Let $\powerset S$ and $\powerset T$ be their power sets.

Let $f \subseteq S \times T$ be a mapping from $S$ to $T$.

Let $f^\gets: \powerset S \to \powerset T$ be the inverse image mapping of $f$:


 * $\forall Y \in \powerset T: \map {f^\gets} Y = \begin {cases} \set {s \in S: \exists t \in Y: \map f s = t} & : \Img f \cap Y \ne \O \\ \O & : \Img f \cap Y = \O \end {cases}$

Then:
 * $\map {f^\gets} T = \Preimg f$

where $\Preimg f$ is the preimage set of $f$.