Suppose $\mathbb{P}$ is a definable proper forcing (for instance Sacks forcing). Let $\alpha$ be some ordinal. Let $\mathbb{P}\_\alpha$ be the countable support iteration of $\mathbb{P}$ of length $\alpha$. 


It is well known that $\mathbb{P}\_\alpha$ is also proper. Hence for any countable elementary structure $M \prec H\_\Theta$ for sufficiently large $\Theta$ such that $\mathbb{P}\_\alpha \in M$ and any $p \in \mathbb{P}\_\alpha \cap M$, there exists some $q \leq\_{\mathbb{P}\_\alpha} p$ which is a $(M, \mathbb{P}\_\alpha)$-generic condition. 


Let $X = \alpha \cap M$. The question is whether a $(M, \mathbb{P}\_\alpha)$ generic condition $q \leq\_{\mathbb{P}\_\alpha} p$ can be found such that $\text{supp}(q) \subseteq X$. 


Thanks for any information that can be provided. 



Iteration of Proper Forcing and Support of Master Conditions
I may be missing something here, because if I've got the question right then the definability doesn't come in to play:


The answer is yes, and the reason is implicit in the proof that $\mathbb{P}\_\alpha$ is proper.
Shelah's proof of Theorem 3.2 on page 109 of Proper and Improper forcing mentions this. In particular, (\*) on the bottom of page 109 points out when building the generic conditions, "we could add $dom(r)\cap [i, j) = N\cap [i, j))$'' which decodes into what you are looking for in the case where $i = 0$ and $j = \alpha$.


