A uniform version of a theorem by Lindström

We prove the following uniform version of a theorem by Lindström: Let \mbox{\cal F}:=\{F_i:~ i\in I\} be a $k$-uniform set family of $[n]$, where $k\ge 1$. If |\mbox{\cal F}|\geq n+1, then there exist two disjoint subsets $I_1$ and $I_2$ of $I$ for which $$\bigcup _{i\in I_1}{M}_i=\bigcup _{i\in I_2}{M}_i$$ and $$\bigcap _{i\in I_1}{M}_i=\bigcap _{i\in I_2}{M}_i\mathrm{.}$$ Our proof uses basic linear algebra.