Let $\mathcal{M} : = \{ 0, 1 \}^X$ denote the
D68: Set of maps from $X$ to $\{ 0, 1 \}$. Result
R1841: Indicator function operator is a bijection shows that
\begin{equation}
|\mathcal{P}(X)|
= |\mathcal{M}|
\end{equation}
and result
R4314: Number of boolean functions on a finite set shows that
\begin{equation}
|\mathcal{M}|
= 2^{|X|}
\end{equation}
$\square$
