Theorem 2.65 in Woodin's book shows that a saturated ideal on $\omega_1$ exists after Levy-collapsing a Woodin cardinal $\delta$ to $\omega_2$. I am confused about the part of the argument where he shows that ideal he defines is a proper ideal.

Claim (2.2) on page 50 says we can find a countable structure $X \prec H_{\delta^+}$ and an $X$-generic condition $p$ such that for all inaccessible $\gamma \in X \cap \delta$ and $Col(\omega_1,<\gamma)$-names $\tau \in X$ for a semi-proper subset of $\mathcal P(\omega_1)/NS$, there is a $\sigma \in X$ such that $p \Vdash \sigma \in \tau$ and $p \Vdash X \cap \omega_1 \in \sigma$. He says we construct this pair $(X,p)$ by an elementary chain.

Suppose $X_0 \prec H_{\delta^+}$ and $p_0 \restriction \gamma \in X_0$. By definition of semi-proper, if $\tau \in X_0$ is as above, then $p_0 \Vdash_\gamma (\exists y \in \tau) Sk(X_0[G_\gamma] \cup \{ y \}) \cap \omega_1 = X_0[G_\gamma] \cap \omega_1$ and $X_0[G_\gamma] \cap \omega_1 \in y$. The problem I have is: How do we choose a *name* for $y$ with a similar property? There is a name $\sigma$ such that $p_0$ forces $\sigma^G$ witnesses the above property of $y$ in $V[G_\gamma]$, but could it be that $Sk(X_0 \cup \{ \sigma \}) \cap \omega_1 \not= X_0 \cap \omega_1$?

Or perhaps there is a quite different strategy for building the elementary chain. Thanks for your help!