TITLE: Axiom of Choice and Right Inverse
QUESTION [20 upvotes]: I read an Theorem that states:

Let $A$ and $B$ be non-empty sets, and let $f:A \to B$ be a function,
  then the function $f$ has a right inverse if and only if $f$ is
  surjective.

The Theorem proof uses Axiom of Choice. My question is, if Axiom of Choice is not true, then is this Theorem false? My question is because I tried to find a proof that not uses the axiom but I did not found it and this result is very important because it implies, with another statement, that a function $f$ has an inverse if and only if $f$ is bijective.

REPLY [4 votes]: Take the subset $E$ in $A\times \mathcal{P}(A)$ defined by $\{(a,B) : a\in B\}$.
Then the second projection is a surjective map onto 
$\mathcal{P}^*(A)=\mathcal{P}(A)\smallsetminus\{\emptyset\}$.
A right inverse, composed with the first projection, yields a selecting function as in AC.