The Clowder Project

\begin{align*} \text{Ran}_{R}\webleft (V\webright )& \cong \int _{a\in A}\textup{Hom}_{\{ \mathsf{t},\mathsf{f}\} }\webleft (R^{-_{2}}_{a},U^{-_{1}}_{a}\webright )\\ & =\webleft\{ b\in B\ \middle |\ \int _{a\in A}\textup{Hom}_{\{ \mathsf{t},\mathsf{f}\} }\webleft (R^{b}_{a},U^{\star }_{a}\webright )=\mathsf{true}\webright\} \\ & = \webleft\{ b\in B\ \middle |\ \begin{aligned} & \text{for each $a\in A$, at least one of the}\\[-2.5pt]& \text{following conditions hold:}\\[7.5pt]& \mspace {25mu}\rlap {\text{1.}}\mspace {22.5mu}\text{We have $R^{b}_{a}=\mathsf{false}$}\\ & \mspace {25mu}\rlap {\text{2.}}\mspace {22.5mu}\text{The following conditions hold:}\\[7.5pt]& \mspace {50mu}\rlap {\text{(a)}}\mspace {30mu}\text{We have $R^{b}_{a}=\mathsf{true}$}\\ & \mspace {50mu}\rlap {\text{(b)}}\mspace {30mu}\text{We have $U^{\star }_{a}=\mathsf{true}$}\\[10pt]\end{aligned} \webright\} \\ & = \webleft\{ b\in B\ \middle |\ \begin{aligned} & \text{for each $a\in A$, at least one of the}\\[-2.5pt]& \text{following conditions hold:}\\[7.5pt]& \mspace {25mu}\rlap {\text{1.}}\mspace {22.5mu}\text{We have $b\not\in R\webleft (A\webright )$}\\ & \mspace {25mu}\rlap {\text{2.}}\mspace {22.5mu}\text{The following conditions hold:}\\[7.5pt]& \mspace {50mu}\rlap {\text{(a)}}\mspace {30mu}\text{We have $b\in R\webleft (a\webright )$}\\ & \mspace {50mu}\rlap {\text{(b)}}\mspace {30mu}\text{We have $a\in U$}\\[10pt]\end{aligned} \webright\} \\ & = \webleft\{ b\in B\ \middle |\ \begin{aligned} & \text{for each $a\in A$, if we have}\\ & \text{$b\in R\webleft (a\webright )$, then $a\in U$}\end{aligned} \webright\} \\ & = \webleft\{ b\in B\ \middle |\ R^{-1}\webleft (b\webright )\subset U\webright\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R^{-1}\webleft (U\webright ).\end{align*}