Functoriality. The assignments $U,V,\webleft (U,V\webright )\mapsto U\cap V$ define functors
\[ \begin{array}{ccc} U\cap -\colon \mkern -15mu & \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ) \mkern -17.5mu& {}\mathbin {\to }\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\\ -\cap V\colon \mkern -15mu & \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ) \mkern -17.5mu& {}\mathbin {\to }\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\\ -_{1}\cap -_{2}\colon \mkern -15mu & \webleft (\mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (X\webright ),\subset \times \subset \webright ) \mkern -17.5mu& {}\mathbin {\to }\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ). \end{array} \]
In particular, the following statements hold for each $U,V,A,B\in \mathcal{P}\webleft (X\webright )$:
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If $U\subset A$, then $U\cap V\subset A\cap V$.
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If $V\subset B$, then $U\cap V\subset U\cap B$.
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If $U\subset A$ and $V\subset B$, then $U\cap V\subset A\cap B$.