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:

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

Proof

 $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle X$$ $$\in$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle \mathcal P \left({S \cap T}\right)$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle \iff$$ $$\displaystyle$$ $$\displaystyle X$$ $$\subseteq$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle S \cap T$$ $$\displaystyle$$ $$\displaystyle$$ Definition of Power Set $$\displaystyle$$ $$\displaystyle \iff$$ $$\displaystyle$$ $$\displaystyle X$$ $$\subseteq$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle S \land X \subseteq T$$ $$\displaystyle$$ $$\displaystyle$$ Definition of intersection $$\displaystyle$$ $$\displaystyle \iff$$ $$\displaystyle$$ $$\displaystyle X$$ $$\in$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle \mathcal P \left({S}\right) \land X \in \mathcal P \left({T}\right)$$ $$\displaystyle$$ $$\displaystyle$$ Definition of Power Set $$\displaystyle$$ $$\displaystyle \iff$$ $$\displaystyle$$ $$\displaystyle X$$ $$\in$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle \mathcal P \left({S}\right) \cap \mathcal P \left({T}\right)$$ $$\displaystyle$$ $$\displaystyle$$ Definition of intersection

$\blacksquare$