We have
\[ \chi _{\mathcal{P}\webleft (X\webright )}\webleft (\chi _{x},\chi _{U}\webright )=\chi _{U}\webleft (x\webright ) \]
for each $x\in X$, giving an equality of functions
\[ \chi _{\mathcal{P}\webleft (X\webright )}\webleft (\chi _{\webleft (-\webright )},\chi _{U}\webright )=\chi _{U}, \]
where
\[ \chi _{\mathcal{P}\webleft (X\webright )}\webleft (U,V\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \mathsf{true}& \text{if $U\subset V$,}\\ \mathsf{false}& \text{otherwise.} \end{cases} \]
