seeking a reference for a result stated by Woodin in "Suitable Extender Models I"

Question

In Section 2.1 of the first part of his two-part paper "Suitable Extender Models" Woodin states the following result as Theorem 5:

Suppose that $\delta$ is a limit of Woodin cardinals and that $T$ is a tree on $\omega\times\kappa$ for some $\kappa$. Then the following are equivalent.

(1) $T$ is $(<\delta)$-weakly homogeneous. (2) There is a tree $S$ on $\omega\times\delta$ such that for all $\mathbb{P}\in V_{\delta}$, if $G\subseteq\mathbb{P}$ is $V$-generic then in $V[G]$,

$$p[T]=\mathbb{R}^{V[G]}\setminus p[S]. $$

This result doesn't seem to be stated in the set of notes "The Stationary Tower". I was wondering if anyone knew a reference for it.

Answer 1

The perfect reference for this is

MR2562557 (2010j:03061) Reviewed. Steel, J. R.(1-CA). The derived model theorem. In Logic Colloquium 2006. Proceedings of Annual European Conference on Logic of the Association for Symbolic Logic held at the Radboud University, Nijmegen, July 27–August 2, 2006, S. B. Cooper, H. Geuvers, A. Pillay and J. Väänänen, eds., Lecture Notes in Logic, 32. Association for Symbolic Logic, Chicago, IL; Cambridge University Press, Cambridge, 2009, pp. 280–327.

A preprint can be accessed from John Steel's website. If you are not already familiar with universally Baire sets, then I also recommend you read the original reference (mentioned by Asaf in a comment), mainly because it is beautiful mathematics, but John's paper is self-contained and addresses this result explicitly.