Given a family $(X_i)_{i \in I}$ of non-empty sets, why cannot one introduce a function $f : I \to \bigcup_{i \in I} X_i$ as follows :
for all $i \in I$, there exists $x \in X_i$ — let $f(i) := x$,
thus producing a "choice function" ?
Why isn't the above construction valid in $\sf {ZF}$ ?