I am interested in better understanding the following property:

Let us say that an iteration of forcings $\langle\mathbb{P}_\alpha,\dot{\mathbb{Q}}_\beta\mid\alpha\leq\gamma,\beta<\gamma\rangle$ is $\kappa$-limited if for all $V$-generic filters $G\subseteq\mathbb{P}_\gamma$ and all $\eta<\kappa$, $$(V^\eta)^{V[G]}=\bigcup_{\alpha<\gamma}(V^\eta)^{V[G\upharpoonright\alpha]}$$ where $G\upharpoonright\alpha$ is the restriction of $G$ to its first $\alpha$ co-ordinates.

I have found that this is equivalent to saying that for all $\eta<\kappa$ and all collections $\{A_\alpha\mid\alpha<\eta\}$ of maximal antichains in $\mathbb{P}_\gamma$, there is a maximal antichain $A\subseteq\mathbb{P}_\gamma$ such that: For all $q\in A$ there is $\delta<\gamma$ such that for all $\alpha<\eta$ and $p\in A_\alpha$, if $p\mathrel{\|}q$ then for all $r_0\leq p\upharpoonright\delta,q\upharpoonright\delta$, $r_0\mathrel{\Vdash}q/\delta\leq p/\delta$.

Here, $p\upharpoonright\delta$ refers to the first $\delta$ co-ordinates of $p$, and $p/\delta$ refers to the $\mathbb{P}_\delta$-name for the final $\gamma\backslash\delta$ co-ordinates of $p$.

In the case of $\mathbb{P}_\gamma$ being a product, we can replace the final clause by "if $p\mathrel{\|}q$ then $q/\delta\leq p/\delta$".

Question: Is the property that I have called $\kappa$-limitedness known in the literature, and if so where can I find information on it? Similarly, is the characterisation by antichains known in the literature?



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