Intersection of Power Sets

Theorem
The intersection of the power sets of two sets $$S$$ and $$T$$ is equal to the power set of their intersection:

$$\mathcal{P} \left({S}\right) \cap \mathcal{P} \left({T}\right) = \mathcal{P} \left({S \cap T}\right)$$

Proof

 * First, we show $$\mathcal{P} \left({S \cap T}\right) \subseteq \mathcal{P} \left({S}\right) \cap \mathcal{P} \left({T}\right)$$:


 * Next, we show $$\mathcal{P} \left({S}\right) \cap \mathcal{P} \left({T}\right) \subseteq \mathcal{P} \left({S \cap T}\right)$$: