The strong inverse image function associated to $R$ is the function
\[ R_{-1}\colon \mathcal{P}\webleft (B\webright )\to \mathcal{P}\webleft (A\webright ) \]
defined by[1]
\[ R_{-1}\webleft (V\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ a\in A\ \middle |\ R\webleft (a\webright )\subset V\webright\} \]
for each $V\in \mathcal{P}\webleft (B\webright )$.
