I'm having a hard time proving the following claim:
Let $j\colon M\rightarrow N$ be an elementary embedding (between inner models) with $\operatorname{crit}(j)=\kappa$. Let $\kappa<\lambda$.
Let $\mu$ be the minimal $\alpha$ with $\lambda\le j(\alpha)$ and let $E=\langle E_a\mid a\in[\lambda]^{<\omega}\rangle$ be the $(\kappa,\lambda)$-extender derived from $j$.
Then, for every $\langle a_n\mid n<\omega\rangle$ (with $a_n\in[\lambda]^{<\omega}$) and $\langle X_n\mid n<\omega\rangle$ (with $X_n\in E_{a_n}$), there is a function $f\colon\bigcup\{a_n\mid n<\omega\}\rightarrow\mu$ such that for each $n<\omega$, we have $f"a_n\in X_n$.
I'm interested in a proof/hint for this claim.
Any help would be appreciated, thanks.