I am reading through the chapter on iterated forcing from Halbeisen's set theory book and have some trouble about the notation that he uses.
Let $\mathbb{P}_{\alpha} = \langle \mathring{\mathbb{Q}}_{\gamma}: \gamma \in \alpha\rangle$ be an $\alpha$-stage iteration and let $G$ be $\mathbb{P}_{\alpha}$-generic over $\mathbf{V}$. Then, for $\beta \in \alpha$, he introduces the following objects:
\begin{equation*} G(\beta) = \{q_{\beta}: \exists \langle \mathring{p}_{\gamma}:\gamma \in \alpha\rangle \in \mathring{G}[q_{\beta} = \mathring{p}_{\beta}[G]]\} \end{equation*}
and \begin{equation*} G\vert_{\beta} = \{\langle q_{\gamma}: \gamma \in \beta\rangle: \exists \langle \mathring{p}_{\gamma}: \gamma \in \alpha\rangle \in \mathring{G}\forall \gamma \in \beta({q}_{\gamma}=\mathring{p}_{\gamma}[G])\}. \end{equation*}
He then claims, $G \vert_{\beta}$ is the $\mathbb{P}_{\beta}$-generic filter generated by $G$. Now, the definition must be wrong since $G \vert_{\beta}$ - as defined - is not even a subset of $\mathbb{P}_{\beta}$. Am I mistaken? If not, what are the correct definitions of $G\vert_{\beta}$ and $G(\beta)$?
