[super_mario]

In ZFC set theory, indexed family over a set (possibly uncountable or even bigger), is just syntactic sugar for a function.

So let's say you have a set U (possibly uncountable). To say let {u_i}, i in I (another set, possibly uncountable) is equivalent to asserting existence of function f:I -> U, such that f(i) = u_i. Note that this does not even require axiom of choice, since you are allowed to postulate that a function exists.

Of course if I is uncountable you can't list the elements of I, but that is not important in this case.