Matching two equal-block partitions by nonempty intersections

Suppose that \(X\) is a set of size \(mn\) and \((A_i)_{i=1}^m\), \((B_i)_{i=1}^m\) are partitions of \(X\) into \(m\) sets of size \(n\). Prove that the sets \(B_i\) can be renumbered so that \[ A_i\cap B_i\ne\varnothing \] for \(i=1,2,\ldots,m\).