Question on how to use the axiom of choice in an extension problem

Question

I am in doubt about the proper use of the axiom of choice I use extension theorem that says that “there exists” a function with a property, call it property A. Then, I am wondering if I can use the axiom of choice to pick all such extensions, my understanding of the axiom of choice is that I can do that, but am I missing something? More detail: I am working on a problem related to the representation of a binary order by means of a collection of utility functions. So I am using a result that guarantees that there exists some utility functions that guarantee that for any 4 points, $x,y,\hat{x},\hat{y}$ in $X$ such that $(x,y)\in R$, and $(\hat{x},\hat{y})\in \lnot R$, where $R$ is the binary relation in question and $\lnot R$ is the complement of $R$ in $X\times X$; there exists a utility function such that $u(x)\geq u(y)$ and $u(\hat{y})>u(\hat{x})$. So that's all good, but then I want to make sure I can select all such extensions, so I need to build a set that contains all such extensions for each $(\hat{x},\hat{y})\in \lnot R$.