The class $L$ of constructible sets is defined by recursion using the operation def$(M)=\{x \subset M: x$ is definable over $(M, \in) \}$. By adding a unary predicate, $P$, to our language, we can similarly define the class $L_{P}$ by using the operation def$_{P}(M)=\{x \subset M: x$ is definable over $(M,\in,P)\}$ instead of def$(M)$.
Assuming that $V \models$ ZFC (with Replacement and Comprehension schemas extended to included formulas that use $P$), is it true that $L_{P}$ is the smallest inner model of $V$ that satisfies these extended Replacement and Comprehension schemas?