Partition of Indexing Set induces Bijection on Family of Sets

Theorem
Let $I$ be an indexing set.

Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a family of sets indexed by $I$.

Let $\family {I_\gamma}_{\gamma \mathop \in J}$ be a partitioning of $I$.

Then there exists a bijection:
 * $\ds \phi: \prod_{\gamma \mathop \in J} \paren {\prod_{\alpha \in \mathop I_\gamma} S_\alpha} \to \prod_{\alpha \mathop \in I} S_\alpha$

Proof
First a lemma: