Hi, and welcome to the forum.

Let where is a domain, is a signature, and is an interpretation of on .

This probably means elements of D that occur in phi.I can't figure it out what the "parameters of phi(x)" are.

What is phi(M)? I'll assume it is where , and is the restriction of on . However, it is not clear that this is a structure, namely, that is closed with respect to functions. (It may happen that for a functional symbol and , .) So, I'll assume that there are no functional symbols in .Prove that if phi(M) contains the parameters of phi(x), then phi(M) is not an elementary substructure of M.

Next, what happens if , e.g., when is ? Then phi(M) = M and so phi(M) is an elementary substructure of M. So, I'll assume that , but , i.e., there exists an such that .

In this case, what about ?

Maybe I made too many assumptions here...