Briefly, my question is this:
Is the following true?
$ZF - Regularity \models \varphi $
where $\varphi$ is the formula in the language of set theory defined as "For all sets $A$ and $B$, if there exists a well-ordering on $A$ and a well ordering on $B$, then there exists a well-ordering on $A^B$", where $A^B$ is the set of all functions from $B$ into $A$. Clearly this statement holds in ZFC, but does it hold for ZF, even without regularity attached?
My main concern is with the question above, so the acceptance of an answer is going to be based solely on the above question. However, if you can point me to a reference about the following harder question, then that would be appreciated: How many axioms can we take away from ZFC and yet $\varphi$ still hold? More precisely, what is the smallest subset $\Gamma \subseteq ZFC$ such that $\Gamma \models \varphi$? Where should I look to find information regarding this "minimalist" type of questions? A good reference textbook on model theory is all I have in mind.
Thank you very much in advance!