Suppose \(A\) is a partitioned set and \(P,Q\) are two partitions such that for each \(p \in P\) there exists a \(q \in Q\) with \(A_p=A_q\)
Show that for each \(p \in P\) there is at most one \(q \in Q\) such that \(A_p = A_q\)
Show that for each \(q \in Q\) there is a \(p \in P\) such that \(A_p = A_q\)
Suppose \(q' \ne q\). If they are both equal to \(A_p\) then they are equal to each other, but a partition rule is that \(q' \ne q\) must have an empty intersection (and \(A_p\) cannot be empty by the other rule).
By part 1, the mapping between part labels is a bijection, so there is an inverse map as well.