I'm self studying the proof of consistency of MA on Jech's Set Theory (Theorem 16.13, p. 272). There is a step which I can't understand. To simplify the notation, I will try to "extract" the relevant fact he uses:
Let $P$ be a forcing poset such that $|P| < \omega_2$. Assume that P has the ccc. Suppose that $\tau \in V^P$ is such that $1_P \Vdash |\tau| \leq \check{\lambda}$ for some $\lambda < \omega_2$. Then every $\sigma \in V^P$ which is an element of $\tau$ can be represented by a function from an antichain in $P$ into $\lambda$ (and thus there are at most $\omega_2$-many).
I don't understand why this is true (but I do understand the "and thus..."). I know that there are nice names (from Kunen's book) and that it's possible to count the number of nice names for subsets of $\tau$ (once we know the size, in $V$, of $|\operatorname{dom}(\tau)|$). I'm also aware of the fact (and, most importantly, the proof) that, when using the boolean valued models approach, we have that $(2^\lambda)^{V[G]} \leq (|B|^\lambda)^V$ (this is Lemma 15.1 on Jech's book).
These results both look close to the topic of my question, but the setting is quite different, so I don't know how to use these facts.
Since $1_P \Vdash |\tau| \leq \check{\lambda}$, there is some name $f$ such that $1_P \Vdash f:\tau \to \check{\lambda}\text{ is an injection}$. In particular, if $p\Vdash\sigma\in\tau$, then we can pick a maximal antichain $A$ of elements of $P$ below $p$ which determine the value of $f(\sigma)$. The function which takes $a\in A$ to the $\alpha<\lambda$ such that $a\Vdash f(\sigma)=\alpha$ then determines $\sigma$.