A uniform version of a theorem by Lindstr\"om

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