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Thank you very much! I have carefully revised the proof and I have found no problem. In fact, $dom(f\cap V_{\kappa_n})\subset V_{\kappa_n}$, and so $\bigcup_{m\in\omega}$ can be replaced by the single term corresponding to $m=n$. A very beautiful idea!
Sorry, I have seen your comment just now. The problem is that $j^+$ is defined on subsets of $V_\lambda$, and $j^+(f_i)$ is just a subset of $V_\lambda$. It will be a total function if the property $\forall x\in V_\lambda^m\exists y\in V_\lambda\ (x, y)\in f_i$ is preserved by $j^+$, but this is a particular case of what we must prove (namely, that $j^+$ preserves a formula with a second order variable).
