\begin{definition}[Definition:Inverse Image Mapping/Mapping/Definition 1]
Let $S$ and $T$ be sets.
Let $\powerset S$ and $\powerset T$ be their power sets.
Let $f: S \to T$ be a mapping.
The '''inverse image mapping''' of $f$ is the mapping $f^\gets: \powerset T \to \powerset S$ that sends a subset $Y \subseteq T$ to its preimage $f^{-1} \paren T$ under $f$:
:$\forall Y \in \powerset T: \map {f^\gets} Y = \begin {cases} \set {s \in S: \exists t \in Y: \map f s = t} & : \Img f \cap Y \ne \O \\ \O & : \Img f \cap Y = \O \end {cases}$
\end{definition}