Axiom of choice, proving a function is onto

Question

I had some questions about the Axiom of choice.

suppose I have a function f:A->B, where A and B are infinite sets, and I have to prove f is onto.

So as a general strategy I pick an arbitrary element b in B, and I find an element a in A, such that f(a) =b.

My question is, is picking an arbitrary element in B making use of the axiom of choice?

Also... seems to me the axiom of choice would not be sufficient, because the way I understand it, the axiom of choice allows us to pick one element out of a set, but not necessarily every element.

But to prove B is onto, I'd have to be able to pick any and every element in B right?

Answer 1

The axiom of choice is not needed. You are choosing one element $b$ from $B$, presumably a non-empty set. This uses one existential instantiation.

Then you prove the set of elements mapped to that chosen element is also non-empty, and you do it by showing there is some $a$ mapped to $b$.

You are not making infinitely many arbitrary choices, so the axiom of choice is not used. Not in "general" anyway.