I am reading the book of Robert Soare for recursive enumerable sets and degrees. There is the so called listing theorem, which stands that a set A is recursive enumerable if and only if A is not empty or A is the range of a total recursive function. In the proof of the $ \Rightarrow $ implication, the function is defined as $$ f(<s, x>) = \begin{cases} x & if \ x \in W_{e, s+1} - W_{e, s} \\ a & otherwise \end{cases} $$ where $<a, t>$ is the least integer such that $ a \in W_{e, t}$
The theorem also says that $f$ can be found uniformly, i.e., $ \exists \psi $ such that it is p.r., $ \lambda y \psi(e, y) $ is total, and $ W_{e} = \{ \psi(e, y) | y \in \omega \} $
The question is, why is not valid a function defined as $ f(x) = \begin{cases} x & if x \in A \\ a & otherwise \end{cases} $
I mean, if computing the conditions on the first definition for $f$ is r.e. it should also be for the second one. And apparently, the $ \psi $ function can be defined as well.
I think that it can not be as easy as I think, so there should be some missconception for my part.