The definability of the extender sequence $\mathbb{E}$ from $\mathbb{E}{\upharpoonright}\aleph_1$ in $L[\mathbb{E}]$

Abstract: Let $M$ be a short extender mouse. We prove that if $E\in M$ and $M$ satisfies "$E$ is a countably complete short extender whose support is a cardinal $\theta$ and $\mathcal{H}\theta\subseteq\mathrm{Ult}(V,E)$", then $E$ is in the extender sequence $\mathbb{E}^M$ of $M$. We also prove other related facts, and use them to establish that if $\kappa$ is an uncountable cardinal of $M$ and $\kappa^{+M}$ exists in $M$ then $(\mathcal{H}{\kappa^+})^M$ satisfies the Axiom of Global Choice. We prove that if $M$ satisfies the Power Set Axiom then $\mathbb{E}^M$ is definable over the universe of $M$ from the parameter $X=\mathbb{E}^M\upharpoonright\aleph_1^M$, and $M$ satisfies "every set is $\mathrm{OD}_{{X}}$". We also prove various local versions of this fact in which $M$ has a largest cardinal, and a version for generic extensions of $M$. As a consequence, for example, the minimal proper class mouse with a Woodin limit of Woodin cardinals models "$V=\mathrm{HOD}$". This adapts to many other similar examples. We also describe a simplified approach to Mitchell-Steel fine structure, which does away with the parameters $u_n$.