On Disjoint Chains of Subsets

We prove the following theorem concerning the poset of all subsets of [n] ordered by inclusion. Consider any two equal-size families of subsets of [n], S and R, where within each family all subsets have the same number of elements. Suppose there exists a bijection φ: S↦R such that A⊃f(A) for all A∈S. Then there exist |S| disjoint saturated chains containing all the subsets in S and R.