The direct image function associated to $R$ is the function
\[ R_{*}\colon \mathcal{P}\webleft (A\webright )\to \mathcal{P}\webleft (B\webright ) \]
\begin{align*} R_{*}\webleft (U\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R\webleft (U\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\bigcup _{a\in U}R\webleft (a\webright )\\ & = \webleft\{ b\in B\ \middle |\ \begin{aligned} & \text{there exists some $a\in U$}\\ & \text{such that $b\in R\webleft (a\webright )$} \end{aligned} \webright\} \end{align*}
for each $U\in \mathcal{P}\webleft (A\webright )$.
1Further Terminology: The set $R\webleft (U\webright )$ is called the direct image of $U$ by $R$.
2We also have
\[ R_{*}\webleft (U\webright )=B\setminus R_{!}\webleft (A\setminus U\webright ); \]
see Item 7 of Proposition 7.4.1.1.3.
