I want two show that a certain property $u$ of some finite structure is not definable in first-order logic.
Is the following reasoning correct?
Let $\mathcal{S}$ denote a finite structure. Further, we can "transform" $\mathcal{S}$ to another finite structure $\mathcal{S'}$ (through some kind of bijection).
We already know that a property $p$ of $\mathcal{S}$ is not definable in first-order logic. Now we assume that $u$ is definable by a sentence $\varphi$. Further, we can show that $ \mathcal{S'} \models \varphi \iff \mathcal{S} \text{ has property } p. $
That this prove that property $u$ of $\mathcal{S}$ of is not definable in first-order logic?