\begin{definition}[Definition:Inverse Image Mapping/Mapping/Definition 2]
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 direct image mapping of the inverse $f^{-1}$ of $f$:
:$f^\gets = \paren {f^{-1} }^\to: \powerset T \to \powerset S$:
That is:
:$\forall Y \in \powerset T: \map {f^\gets} Y = \set {s \in S: \exists t \in Y: \map f s = t}$
\end{definition}