We can state the Axiom of Choice as follows: 'If A is a family of nonempty sets, then there is a function f with domain A such that f(a) ∈ a for every a ∈ A. Such a function f is called a choice function for A'.
However, if A may be any family of nonempty sets, it may also include the set that we create through the choice of exactly one element from a known collection of nonempty sets. That is, if we allow any set to exist in the first place, we either do not need a choice function to create a new set, or we tacitly assume that at the beginning the choice function only applies to certain collections of sets. But since the Choice Set is a set, then after its statement of existence we may apply the choice function to the collection including this set and other sets to create a new set that did not exist before.
Am I wrong? My problem is: which sets are allowed to exist before the statement of existence of the Choice set?