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Suppose that $\mathscr{H}$ is a set such that each element $H\in \mathscr{H}$ is a set. Prove the following: $\bigcap\limits_{H\in \mathscr{H}}\mathscr{P}(H)= \mathscr{P}(\bigcap\limits_{H\in \mathscr{H}}H)$.

I know that there a lot of people who have asked the same question, but it's for the specific case of two sets. I am stuck on trying to prove the general case.

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    The proof is the same as the case of two. Just use the definition of power sets and of intersection. – 2017-02-23
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    Also, you need that $\scr H$ is not empty. – 2017-02-23

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