Disjoint subsets problem complexity Is the decision problem below NP-complete?

Given sets $S_1, ... , S_n$, as well as bounds $b_1, ... , b_n$, is it possible to pick pairwise disjoint subsets $U_1, ... , U_n$ such that $U_i \subset S_i$ and $|U_i| \geqslant b_i$ for all $i$?
Create a bipartite graph:


For every element $x\in\bigcup_{i=1}^nS_i$, introduce a corresponding vertex $u(x)$.
For every set $S_i$, introduce a corresponding vertex $v(S_i)$.
Connect vertex $u(x)$ to vertex $v(S_i)$ by an edge, if and only if $x\in S_i$.


Then your problem essentially asks, whether there exists a subset $F$ of the edges, so that


every vertex $u(x)$ is incident to at most one edge in $F$, and
every vertex $v(S_i)$ is incident to at least $b_i$ edges in $F$.


This is an instance of the so-called $f$-factor problem, and hence solvable in polynomial time.
See for instance the book "Matching Theory" by László Lovász and Michael D. Plummer.