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?
Axioms do not create sets. Axioms let us prove that some objects, in this case sets, exist. But existence is not predicated on provability.
Some families of sets admits a choice function, and that much is provable without the axiom of choice. If a family does not admit a choice function, then sets from which you can construct a choice function also do not exist.