Power-set axiom in the constructible universe (${\mathcal{P}}^L(a)={\mathcal{P}}^V(a)\cap L\in L$)

For $a\in L$, ambient separation forms $S=\mathcal{P}(a)\cap L$. Ambient replacement bounds the first constructible stages of all members of $S$. At a stage containing $a$ and all of $S$, the latter is definable by the bounded property of being a subset of $a$. Thus $S$ belongs to the next constructible stage and is the internal power set in $L$.