I try to prove the Jech's "Set theory", exercises 12.11:
12.11. If $\kappa$ is an inaccessible cardinal, then $V_\kappa\models \text{there is a countable model of ZFC}$.
My attempt. Since $(V_\kappa,\in)$ is a model of ZFC, by Löwenheim-Skolem theorem theorem there is a countable model $(A,R)$ which elementary equivalent to $(V_\kappa,\in)$. Especially, $(A,R)$ is a model of ZFC. Since $A$ is countable, we can find $E\subset \omega\times\omega$ such that $(A,E)$ is isomorphic to $(\omega,E)$. Since $E\in P(\omega\times\omega)$ and $P(\omega\times\omega)\in V_{\omega+\omega}$, so $(\omega,E)\in V_\kappa$.
We will prove that $V_\kappa\models (\omega,E)\text{ is a countable model of ZFC}.$ Since $\omega^{V_\kappa}=\omega$, $V_\kappa$ satisfies "$(\omega,E)$ is a countable structure". To prove $V_\kappa\models ((\omega,E)\models ZFC)$, we will check that $V_\kappa \models \varphi^{\omega,E}$ holds for each axiom $\varphi$ of ZFC. By induction for $\varphi$, we can prove $(\varphi^{\omega,E})^{V_\kappa}\leftrightarrow \varphi^{\omega,E}$. If $\varphi$ is an axiom of ZFC, then $\varphi^{\omega,E}$ holds because $(\omega,E)$ is a model of ZFC, so $(\varphi^{\omega,E})^{V_\kappa}$ holds for each axiom $\varphi$ of ZFC.
My "proof" is correct? If not, how to improve my "proof"? Thanks for any help.
Add : As I think, last part of this "proof" has an error. But I don't know what is wrong exactly.