I am wondering something about definability : Suppose we have an infinite set of finite structures $\mathcal{A}^i$ such that $\forall i \geq 0, \mathcal{A}^i \subseteq \mathcal{A}^{i+1}$, i.e for each $i \geq 0, $ $\mathcal{A}^i$ is a substructure of $\mathcal{A}^{i+1}$. Suppose that I can define a set $S_i$ in each $\mathcal{A}^i$ by a formula of first order logic $\varphi$ such that $S_i= \{ \vec{a} \in A^i, \mathcal{A}^i \vDash \varphi(\vec{a}) \} \subseteq \{ \vec{a} \in A^{i+1}, \mathcal{A}^{i+1} \vDash \varphi(\vec{a}) \} =S_{i+1}$, then can I find a first order formula defining the set $\bigcup \{ \vec{a} \in A^i, \mathcal{A}^i \vDash \varphi(\vec{a}) \} =\bigcup S_i$ in the structure $\mathcal{A}=\bigcup \mathcal{A}^i $?

P.S: Note that $|\mathcal{A}^i| < \omega$ and $|\mathcal{A}|=\omega$.