Let $\mathbb{P}$ and $\mathbb{Q}$ be two forcing notions and $\pi:\mathbb{P}\rightarrow\mathbb{Q}$ be a projection.
Let $G\subset\mathbb{P}$ and $H\subseteq\mathbb{Q}$ the filter generated by $\pi''G$. Define
$$ \mathbb{R} = \left\{p\in \mathbb{P}\ \bigg|\ \pi(p)\in H\right\} $$
then $G\subseteq \mathbb{R}$ and $G$ is $M[H]$-generic filter (in $\mathbb{R}$).
This seems rather trivial, but the move into $M[H]$ can add more dense sets making $G$ not generic, this also means that I cant count on $H$ to remain generic in $M[H]$ (else I would just look at $\pi''\left(D\cup(\mathbb{P}\setminus\mathbb{R})\right)$ which will be dense in $\mathbb{Q}$.
Any help would be appriciated.