Power Set Preserves Intersection
Theorem
For any sets \(A\) and \(B\),
\[ \mathcal{P}(A \cap B) = \mathcal{P}(A) \cap \mathcal{P}(B).\]
Proof
\[\begin{align*}
x \in \mathcal{P}(A \cap B) &\iff x \subseteq A \cap B \\
&\iff x \subseteq A \quad \text{and} \quad x \subseteq B \\
&\iff x \in \mathcal{P}(A) \quad \text{and} \quad x \in \mathcal{P}(B) \\
&\iff x \in \mathcal{P}(A) \cap \mathcal{P}(B) \\
\end{align*}\]