Let $M$ be a c.t.m. of ZFC. Let $\mathbb{P}$ be the poset $Fn(\lambda \times \kappa, 2, \kappa) = \{p : |p| < \kappa \wedge p$ is a function $\wedge dom(p) \subseteq \lambda\times\kappa \wedge ran(p) \subseteq 2\}$, where $\lambda \leq \kappa$ and $\kappa$ regular in $M$. Let $G$ be a $\mathbb{P}$-generic over $M$. We know that this way in $M[G]$ we have $\lambda$ new subsets of $\kappa$. Do we know that everything of rank smaller than $\kappa$ remains unchanged?
I know that since $\kappa$ is regular, we have that $(\mathbb{P}$ is $\kappa$-closed$)^M$, but does this fact give us $$x \in M[G] \text{ and } (rank(x) < \kappa)^{M[G]} \Rightarrow x \in M.$$ If so, then do we also get that $(V_\kappa)^M = (V_\kappa)^{M[G]}$?
This is certainly false if $\kappa$ is not inaccessible.
Consider this forcing when $\lambda=1$ and $\kappa=\aleph_1$. We add one new generic subset to $\omega_1$ without adding countable subsets to the universe.
Since $\aleph_1\leq2^{\aleph_0}$ we have that $|\omega_1|\leq|V_{\omega+1}|$ and therefore we can find an element of $V_{\omega+2}$ which encodes the generic set, but its rank is certainly countable.