Making a choice function if $A_i$ are well-ordered for each $i$

Question

Let $A_i$ be a family of sets such that each $A_i$ is well-ordered. Let $\varphi(x,S,W)$ be the formula $$ \forall z ( (z,x) \in W \rightarrow z \notin S)$$ where $W$ is the well-order on $S$. Then $\varphi(x,S,W)$ is true if and only if $x$ is the $W$-minimal element of $S$.

Let's use Separation to obtain the family $A_i' = \{ x : \varphi(x,A_i,W_i) \} = \{a_i\}$ consisting of one element sets. Next let's make ordered pairs $(i,a_i)$ from these using Pairing. Finally, let's apply Union to pack these pairs into one set $F = \{(i,a_i)\}_{i\in I}$. So far we have not invoked the axiom of choice anywhere. But $F$ seems to be a function $F: I \to \{A_i\}_i$, choosing one element from each $A_i$.

Where is the flaw in my construction? As I understand it's only possible to construct a choice function for $A_i$ if not only each $A_i$ is well-ordered but $\bigcup_i A_i$ is well-ordered, too.

Thanks for your help.

Answer 1

Your flaw is by the fact that you choose $W_i$ for each $A_i$.

It is possible that you cannot make such choice. For example if $\{A_i \mid i\in\omega\}$ is a set of pairs without a choice function, then you cannot choose $W_i$ for all $i$.

However if you have a well-order of $\bigcup A_i$ and you just take $W_i$ as the restriction of this well-order to $A_i$, then you did not use the axiom of choice anywhere. Alternatively if $(A_i,W_i)$ are given to us then you have essentially a uniform way of choosing a well-ordering $W_i$ (simply choose the one given to you).