Shewing the existence of a function assuming V=L

Question

Assume V=L.

Does there exist a function $f : \omega_1 \to \omega_1$ such that

1. $f$ is $\Delta_1$ definable over $L_{\omega_1}$
2. For every $A \subset \omega_1$ and every $\Sigma_1$ formula $\phi$, there is a club set $C \subset \omega_1$ so that for every $\alpha \in C$, if $L \models \phi[A]$ then $L_{f(\alpha)} \models \phi[A \cap \alpha]$