\begin{definition}[Definition:Image of Subset under Mapping/Definition 2]
Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping.
Let $X \subseteq S$ be a subset of $S$.
The '''image of $X$ under $f$''' is the element of the codomain of the direct image mapping $f^\to: \powerset S \to \powerset T$ of $f$:
:$\forall X \in \powerset S: \map {f^\to} X := \set {t \in T: \exists s \in X: \map f s = t}$
\end{definition}