I read something that seems to suggest the following is true:
If $\mathbb{P}$ is a proper forcing, $|\mathbb{P}| = \aleph_1$, and $\mathsf{CH}$ holds, then there exists a sequence $\{(p_\xi, \tau_\xi) : \xi < \omega_1\}$ with $p_\alpha \in \mathbb{P}$ and $\tau_\xi$ a $\mathbb{P}$-name such that for all $p \in \mathbb{P}$ and names $\tau$, if $p \Vdash \tau \in 2^\omega$, then there exists some $\xi < \omega_1$ with $p_\xi \leq p$ and $p_\xi \Vdash \tau = \tau_\xi$.
Is statement indeed true? If so, can someone provide a reference or brief sketch of the proof.
I believe the above should imply that if the ground model satisfy $\mathsf{CH}$, then $\mathsf{CH}$ remains true in a generic extension by a $\aleph_1$ size proper forcing.
Thanks.
This was answered positively by Komjath here under the weaker assumption that $P$ preserves $\omega_1$.