= Solution
For every $x\in a$, the assumption $a\subseteq V$ supplies an <ordinal> stage containing $x$. Let $\alpha_x$ be the least such <ordinal>. Leastness makes this a uniquely definable assignment, so ambient <replacement> gives the <set>
$$
S=\{\alpha_x:x\in a\}.
$$
Put $\beta=\sup S$, taking $\beta=0$ if $a$ is empty. Monotonicity of the hierarchy gives $x\in V_{\alpha_x}\subseteq V_\beta$ for every $x\in a$. Hence $a\subseteq V_\beta$, and therefore
$$
\boxed{a\in\mathcal P(V_\beta)=V_{\beta+1}\subseteq V.}
$$
This is a rank-bounding argument: a set-sized collection of elements from the hierarchy has its stages bounded by one <ordinal>. No simultaneous arbitrary choices are required, since each least stage is uniquely specified.
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