To define a sub-interpretation of of domain it is necessary that for any constant symbol and any -placed function symbol of : and
So what you can do is, given (by the way isn't it ?), you can define the substructure of generated by , whose domain is:
and then for every respectively -ary relation symbol, -ary function symbol and constant symbol of define:
To be able to say that is at most countable, the language has to be at most countable.
But what do you want to prove, the downward L÷wenheim-Skolem theorem?