I am trying to understand the proof of Corollary 18.34 in Jech's Set Theory:
If $0^\sharp$ does not exist, then if $\kappa$ is a singular cardinal and if there exists a nonconstructible subset of $\kappa$, then some $\alpha < \kappa$ has a nonconstructible subset.
In the course of the proof, he assumes that every $\alpha < \kappa$ has only constructible subsets. He then wants to show that it suffices to show that all subsets of $\kappa$ of cardinality $cf(\kappa)$ are constructible. In order show this, suppose $A \subseteq \kappa$ and $(a_\nu : \nu < cf(\kappa))$ is a cofinal sequence. By the assumption $A \cap a_\nu$ is constructible. Jech then claims that $\mathcal{A} = \{A \cap a_\nu : \nu < cf(\kappa)\}$ is a constructible set.
Why is this set constructible? His reasoning is : "$\mathcal{A}$ is a subset of $L_\kappa$ of size $\leq cf(\kappa)$ and hence constructible". I do not see why the constructibility of $\mathcal{A}$ follows from this.
He goes to explain that in the next paragraph.
First note that we can encode this set as a set of ordinals of size $\leq\operatorname{cf}(\kappa)$, such that the encoding and decoding functions are in $L$. Therefore by the covering lemma (which holds in the absence of $0^\#$) there is a set of ordinals covering this set, whose cardinality is still $<\kappa$ (it is $\operatorname{cf}(\kappa)+\aleph_1$), $Y$.
In $L$ we can find a bijection between $Y$ and some $\alpha<\kappa$ (e.g. the Mostowski collapse), and by the assumption all the subsets of $\alpha$ which are in $V$ are in fact in $L$. In particular the subset encoding the original set of ordinals; therefore its decoding - the sequence of initial segments of $A$ - is also in $L$, as wanted.