Use the axioms of ZFC to prove the cartesian product A x B is a set I'm given that x, y, and A, B are sets, and must prove that the Cartesian product 
A x B := {(x, y) | x $\in$ A and y $\in$ B} is a set.
I'm certain that I need to use AC somehow, but I'm having trouble understanding what exactly it would even mean for me to apply it, or how I should go about it. 
Here is what believe to understand about AC: if I have a set, say Z = {a, b, c} then applying AC means that I have a function:
f:Z $\mapsto$ $\cup$Z, i.e., f:{a, b, c} $\mapsto$ {{a}, {b}, {c}} 
And we have that $\forall$z $\in$ Z (f(z) $\in$ z)
Unfortunately I'm having trouble seeing the consequences of this and how I can extend it to prove that the Cartesian product between A and B is indeed a set. But it could be as well that I've totally misinterpreted the axiom and so obviously can't draw any meaningful conclusions. I would really appreciate if someone could clear the air for me and hopefully set me in the right direction.

Use the definition of a pair along with Comprehension. 
No need for Choice.
