# When does an iteration not add functions $\eta\to V$ at the final stage?

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?