Forcing Over CH

Let $\mathbb P = Fn(\omega_2\times\omega,2)$ be the collection of all the finite partial functions from $\omega_2\times \omega$ to $2$. Our strategy is:

• to firstly find a collection of dense sets $D_{\alpha\beta}$, such that a generic filter $G$ can be build upon;
• to secondly prove that any generic filter of $\mathbb P$ preserves cardinals.

Lemma. 1 $D_{\alpha\beta}$ are dense sets, where
$$D_{\alpha\beta} = {p\in\mathbb P\mid \exists n\in\omega(\langle\alpha,n\rangle\in dom(p), \langle\beta,n\rangle\in dom(p),p(\alpha,n)\neq p(\beta,n))}.$$