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Let $M$ be an iterable fine structural mouse.  We prove that if $E\in M$ and $M\models$``$E$ is a countably complete shortextender whose support is a cardinal $\theta$ and$\mathcal{H}_\theta\subseteq\mathrm{Ult}(V,E)$'' then $E$ is in the extendersequence $\mathbb{E}^M$ of $M$. We also prove other related facts, and use themto establish that if $\kappa$ is an uncountable cardinal of $M$ and$(\kappa^+)^M$ exists in $M$ then $(\mathcal{H}_{\kappa^+})^M$ satisfies theAxiom of Global Choice.  We then 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 therefore $M$ satisfies``Every set is $\mathrm{OD}_X$''. We also prove various local versions of thisfact in which $M$ has a largest cardinal, and a version for generic extensionsof $M$.  As a consequence, for example, the minimal proper class mouse with a Woodinlimit of Woodin cardinals models ``$V=\mathrm{HOD}$''. This adapts to manyother similar examples.  We also describe a simplified approach to Mitchell-Steel fine structure,which does away with the parameters $u_n$.

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