Proving that a certain set in a generic forcing extension was already in the ground model

Proving that a certain set in a generic forcing extension was already in
the ground model

Suppose $M[G]$ is a generic extension of a model $M$. Consider a cardinal
$\lambda$ with a partial order $\leq\in M[G]$ and let $\mathscr D$ be a
collection of $\lambda$ dense sets of $\langle\lambda, \leq\rangle$. In
particular, let $\mathscr D=\{D_\alpha:\alpha\in\lambda \}$. Suppose that
the set $X=\{\langle\alpha, \beta \rangle\}\in
\lambda\times\lambda:\beta\in D_\alpha\}$ is already in $M$.
How can I define $\mathscr D$ in terms of $X$ in such a way that it will
imply $\mathscr D$ is also in $M$?
Just for reference, it is related to the proof of the consistency of
Martin's Axiom as it appears in Kunen's book.