Let \(A _ { 1 } , A _ { 2 } , \ldots , A _ { n }\) be finite sets, and \(d _ { 1 } , d _ { 2 } , \ldots , d _ { n }\) non-negative integers. Prove that there are disjoint subsets \(D _ { i } \subset A _ { i }\) with \(| D _ { i } | = d _ { i }\) if and only if for all \(I \subset\) \(\{ 1 , 2 , \ldots , n \}\) we have
\[ \left| \bigcup _ { i \in I } A _ { i } \right| \geq \sum _ { i \in I } d _ { i } . \]