The 'hereditarily countable names' are as defined in Shelah's Proper and Improper Forcing, Chapter 3 Definition 4.1. Let $\mathbb{P}$ be a proper forcing notion and $\dot{Q}$ a $\mathbb{P}$-name such that $\Vdash_{\mathbb{P}}$ "$\dot{Q}$ is a proper forcing notion with set of elements $\check{\kappa}$ and maximal element $\check{0}$".

Let $A$ be the closure of {$\check{\alpha}$ : $\alpha < \kappa$} under the following 2 functions.

(1) given sequences ($p_n$ : $n \in \omega$) and ($\tau_n$ : $n \in \omega$), let $\tau$ be a name forced to be equal to: (i) $\tau_i$ where $i$ is the least $n$ satisfying $p_n \in \dot{G_\mathbb{P}}$, if such $i$ exists, and (ii) $\check{0}$, otherwise.

(2) given ($\tau_{m, n}$ : $m$, $n \in \omega$), let $\tau$ be: (i) the $\epsilon$-least element of $\dot{Q}$ such that for all $m \in \omega$, {$\tau_{m, n}$ : $n \in \omega$} is predense below $\tau$, if such an element exists, and (ii) $\check{0}$, otherwise.

My question is, is it true that if $p \in \mathbb{P}$ and $\sigma$ is a $\mathbb{P}$-name such that $p \Vdash \sigma \in \dot{Q}$, then there is a $\mathbb{P}$-name $\tau \in A$ such that $p \Vdash \tau \le \sigma$, or even better, $p \Vdash \tau = \sigma$? If so, why is that?

Much thanks in advance.