The direct image function associated to $f$ is the function
\[ f_{*}\colon \mathcal{P}\webleft (A\webright )\to \mathcal{P}\webleft (B\webright ) \]
\begin{align*} f_{*}\webleft (U\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f\webleft (U\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ b\in B\ \middle |\ \begin{aligned} & \text{there exists some $a\in U$}\\ & \text{such that $b=f\webleft (a\webright )$} \end{aligned} \webright\} \\ & = \webleft\{ f\webleft (a\webright )\in B\ \middle |\ a\in U\webright\} \end{align*}
for each $U\in \mathcal{P}\webleft (A\webright )$.
