Doing inverse limit reflection from an $I_0$ embedding to an $I_1$ embedding

Question

Suppose that $j:L(V_{\lambda+1}) \rightarrow L(V_{\lambda+1})$ with critical point $\kappa<\lambda$. Then $\kappa$ is a limit of cardinals $\kappa'<\kappa$ such that $I_1(\kappa',\delta)$ for a $\delta<\kappa$, I believe. I'm trying to get clear about whether any form of the axiom of choice is needed to prove this and if so how it comes up.