I am reading Tent, Ziegler "A course in model theory". There is written:
6.1.10 Let $\mathbb{D}$ be a definable class and $A$ a set of parameters. Then$\mathbb{D}$ is definable over $A$ if and only if $\mathbb{D}$ is invariant under all automorphisms of $\mathfrak{C}$ which fix $A$ pointwise.
8.3.3 (Separation of variables) Let T be stable and let $\mathbb{F}$ be a 0-definable class. Then any definable subclass of $\mathbb{F}^n$ is definable using parameters from $\mathbb{F}$. (In this case $\mathbb{F}$ is called stably embedded. In other words 8.3.3 says that in stable theories every 0-definable class is stably embedded.)
In 8.3.3, why do we need T to be stable? Is this statement not trivial with 6.1.10 in mind? (Since every definable subclass of $\mathbb{F}^n$ is invariant under all automorphisms of $\mathfrak{C}$ which fix $\mathbb{F}$ pointwise.)
Regards, Patrice
In 6.1.10, it is very important that $A$ is a (small) set of parameters. It is instructive to work through the proof of 6.1.10 assuming that $A$ is a definable class $\mathbb{F}$ and find exactly what goes wrong.