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The definable power set $\mathcal{D}(A)$ performs one definability step over the single structure $(A,\in )$, whereas the constructible power set ${\mathcal{P}}^L(A)$ contains subsets of $A$ created at arbitrarily late stages of the constructible hierarchy.

For the concrete case $A=\omega$, there are only countably many first-order formulas and finite tuples of natural-number parameters, so $\mathcal{D}(\omega )$ is a countable set. In contrast, the constructible universe satisfies ZFC, and Cantor theorem makes its full power set ${\mathcal{P}}^L(\omega )$ uncountable inside $L$. Consequentlyso the two notions do not agree in general.

Fix $a\in L$. Inside the ambient transitive model $M$, the Axiom of power set makes the collection of constructible subsets of $a$ a set. For each such subset $x$, choose the least stage of the constructible hierarchy at which $x$ appears. The Axiom schema of replacement and the supremum of a set of ordinals give an ordinal $\beta$ bounding all these stages; enlarge $\beta$ so that $a\in L_{\beta }$.

Nowis definable over $L_{\beta }$ with parameter $a$. It therefore belongs to the definable power set $\mathcal{D}(L_{\beta })=L_{\beta +1}\subseteq L$. This set contains exactly the subsets of $a$ that belong to the constructible universe, so it witnesses the Axiom of power set in $L$. Therefore $L\models$ Power Set.