Let $E$ be an infinite set of cardinality $\aleph_\alpha$, and let $\mathfrak{F}\subset\mathfrak{P}(E)$ such that $\bigcup\mathfrak{F}=E$, $|\mathfrak{F}|=\aleph_\alpha$, and $|A|=\aleph_\alpha$ for any $A\in\mathfrak{F}$. Furthermore, I have a mapping $f$ of $\omega_\alpha$ onto $\mathfrak{F}$ such that $|f^{-1}(A)|=\aleph_\alpha$ for any $A\in\mathfrak{F}$. I would like to define a bijection $g$ of $\omega_\alpha$ onto $E$ such that $g(\xi)\in f(\xi)$. (Maybe not all assumptions are necessary.)
How can this be done? It is easy to define an injection $g$ of that form by transfinite induction, but how can I make sure that "no element of $E$ is left after the last step". I had the idea of starting with an arbitrary bijection of $\omega_\alpha$ onto $E$, and only modifying it at each step of the induction (e.g. by exchanging pairs of elements). But I really don't know what to do at the limit ordinal steps. Any kind of help is welcome.
It would also be great for intuition if someone could prove the $\alpha=0$ case.