How does the power set relate to an intersection?
First, if we have a set of sets—denote it
$\mathcal{C} $—and all members are subsets of a
fixed set—denote it $E$—then the set of sets is
a subset of $\powerset{E}$.
In this case, we can write
\[
\bigcap \Set{X \in \powerset{E}}{x \in \mathcal{C} }
\] \[
\bigcap_{X \in \mathcal{C} } X.
\]
Here are some basic interactions between the powerset and intersections.1