Consider the product forcing $\prod_n\text{Fn}_{\aleph_n}(\aleph_n,2)$ with full support, where $\text{Fn}_{\aleph_n}(\aleph_n,2)$ is the poset of partial functions from $\aleph_n$ to $2$ of size less than $\aleph_n$, aka $\text{Add}(\aleph_n,1)$. On one hand, this product is a special case of Easton forcing, so assuming GCH in ground model it doesn't collapse anything. On the other hand, $\text{Fn}_{\aleph_n}(\aleph_n,2)$ is isomorphic to $\text{Fn}_{\aleph_n}(\aleph_n\times\aleph_{n-1},2)$, which has $\text{Fn}_{\aleph_n}(\aleph_n,2^{\aleph_{n-1}})=\text{Fn}_{\aleph_n}(\aleph_n,\aleph_n)$ as a dense subset. Doesn't $\prod_n\text{Fn}_{\aleph_n}(\aleph_n,\aleph_n)$ collapse $\aleph_{\omega}^{\aleph_0}$ to $\aleph_\omega$, since if $f_n:\aleph_n\rightarrow\aleph_n$ is the $n$-th generic map, then $F:\aleph_\omega\rightarrow[\aleph_\omega]^{\aleph_0},\alpha\mapsto\{f_n(\alpha)\}_n$ is surjective? Where did I go wrong?
More generally, is there a "collapse analogue" of Easton's theorem that tells us which patterns of collapse are possible between ground model and generic extension?
Edit: The density argument for surjectivity I had in mind was flawed. So let me change my question: if $\kappa_n$ are increasing with limit $\kappa$, $\mathbb{P}_n$ is a poset of size (around) $\kappa_n$, what are some simple criteria to see whether $\prod_n\mathbb{P}_n$ collapses $\kappa^+$ or not?