prove that any two isomorphic structures are elementarily equivalent

Question

Imagine we have two L-structures $M$ and $N$. For each L-sentence $\phi$ , $M$ models $\phi$ iff $N$ models $\phi$. We call $M$ and $N$ two elementary equivalent L-structures.

We say $M$ and $N$ are isomorphic if there exists a function from $U_M$ to $U_N$ which is 1 to 1 and onto and preserves symbols. ( i mean an embedding which is also onto )

Now i want to prove that if two structures are isomorphic, then they are elementary equivalent.

Note : When we speak about L-formulas, we prove these kind of questions by induction on the complexity of the formulas. but in this case, we have sentences not formulas.

Answer 1

Choose an isomorphism $f$. Prove the more general statement: for any formula $\phi$, if we consider any assignment of values in $M$ to the variables, and the corresponding (via $f$) assignment in $N$, the formula $\phi$ will be satisfied by the one assignment if and only if it is satisfied by the other.

Answer 2

Your notion of isomorphism is incomplete and does not suffice to prove the property you're after.

For the structures $M$ and $N$ to be "isomorphic" in the usual sense, you need not only a bijection between the universes, but the bijection also needs to preserve the meanings of function and predicate symbols.