how to collapse $\omega_2$ to a smaller cardinal

Question

Let $M$ be a model of ZFC and take the forcing notion $P(\omega,\omega_2)$ where: $P(\omega,\omega_2)=\{p|p \space is \space a \space function \space and \space \exists n \space s.t. (dom(p)=n) \space n \space and \space ran(p)\subset \omega_2 \}$. ($G$ is a $P$-generic filter)

I am trying to show that in $M[G]$ we can collapse $\omega_2$ to a smaller cardinal.

Any idea how this can be done?

Thank you

EDIT

Answer 1

For each $\alpha<\omega^M_2$, it is easy to show that $\{p\in P: \alpha\in rng(p)\}\in M$ is dense in $P$. So $rng(\cup G) = \omega^M_2$ and $M[G]\vDash \omega = |\omega^M_2|$.