Bijection between Power Set of Disjoint Union and Cartesian Product of Power Sets

Theorem
Let $S$ and $T$ be disjoint sets.

Let $\powerset S$ denote the power set of $S$.

Then there exists a bijection from $\powerset {S \cup T}$ to $\paren {\powerset S} \times \paren {\powerset T}$.

Proof
Let $\phi: \powerset {S \cup T} \to \paren {\powerset S} \times \paren {\powerset T}$ be defined as: