First, I assume that " is a family of sets with domain " means that is a family indexed by elements of , i.e., for each , is a set.
Each is a set of indices. For example, let and let , and . Then is the set of all indices.
Now suppose that for each index there is a set . In the equation , the left-hand side is something like , if is considered left-associative. The right-hand side is . The original associative law is obtained by taking and .