I want to prove the following:
Let $G$ be a generic subset of $\mathbb{P}=\text{Fn}(\mu,2)$ where $\mu$ is an ordinal, then for any ordinal $\alpha$ there exists $\beta\geq\alpha$ such that for any $x\in V[G]_\beta$ there is a name $\tau\in V_\beta$ such that $\tau_G=x.$
($\text{Fn}(\mu,2)$ is the set of finite partial functions $\mu\rightarrow 2$, i.e., Cohen forcing.)
As an attempt I define $\phi:V[G]\rightarrow\text{On}$ by $$\phi(x) = \min\{\text{rk}(\tau)\mid \tau\in V^\mathbb{P}\text{ and }\tau_G=x\} $$ and $\psi:\text{On}\rightarrow\text{On}$ by $$ \psi(\xi)=\sup\{\phi(x)+1\mid x\in V[G]_\xi\}. $$ Now, a fixed point of $\psi$ would give use the $\beta$ we are looking for. And a way to find a fixed point of $\psi$ would be to show it is continuous and strictly increasing. It is easily verified to be continuous, but I can't show it is increasing and it actually seems to me that it's probably not even true. Thus, the natural thing to do would be to modify $\psi$ slightly so that its fixed points still give us what we want but such that $\psi$ is strictly increasing. One such modification might be: $$\psi(\xi)=\sup\{\phi(x)+\sup_{\alpha<\xi}\psi(\alpha)\mid x\in V[G]_\xi\}.$$ But I can't quite get it to work––it isn't neither clearly increasing nor clearly continuous.
This is an attempt to answer, at least partially, this question.
Thank you for any help!
The argument you used (link) works.
Let $G$ be $\mathbb{P}$-generic over $V$. Fix $\theta_{0} \in OR$ and consider
\begin{gather*} \phi :V[G] \longrightarrow OR \end{gather*} where \begin{gather*} \phi(x) = min\{ \beta \ | \ \exists \sigma \in (V_{\beta})^{V} ( \sigma_{G} = x ) \} \end{gather*} and $\psi: OR \rightarrow OR$, where \begin{gather*} \psi(\alpha) = sup\{ \phi(x) \ | \ x \in V_{\alpha}^{V[G]} \} \end{gather*} Note that for all $\alpha$ we have $\psi(\alpha) \geq \alpha $. Define $\langle \theta_{n} \ | \ n \leq \omega \rangle $, as follows: \begin{gather*} \theta_{n+1} = \psi(\theta_{n}) + 1 \end{gather*} and \begin{gather*} \theta_{\omega} = sup_{n\in\omega} \theta_{n} \end{gather*}
It follows that $ n < m \leq \omega $ implies $\theta_{n} < \theta_{m}$.
(For $\beta \in OR$ we write $V_{\beta}[G]:= \{ \tau_{G} \ | \ \tau \in V_{\beta} \ \& \ \tau \text{ is a name} \}$.) Fix $x \in V_{\theta_{\omega}}^{V[G]}$. Then $x \in V_{\theta_{n}}^{V[G]}$ for some $n < \omega$ and $V_{\theta_{n}}^{V[G]} \subseteq V_{\theta_{n+1}}[G] \subseteq V_{\theta_{\omega}}[G]$.