I'm trying to show that if we have a structure $\textbf{A}$ and $C\subseteq A$, with an $\mathcal{L}_C$ formula $\varphi(x)$ defining the set $S$ in $\textbf{A}$, then for any given automorphism $h\in Aut(\textbf{A})$, we have the $\mathcal{L}_{h(c)}$ formula $h(\varphi)(x)$ defining the set $h(S)$.
The proof seems rather straightforward. By assumption, we have $\textbf{A}\vDash\varphi(\textbf{a})$, for each $a\in S$. So since $h$ is iso we have $\textbf{A}\vDash h(\varphi)(\textbf{h(a)})$ for each $h(a)\in h(S)$. Now consider arbitrary $a\in A$ such that $\textbf{A}\vDash h(\varphi)(\textbf{a})$. Since each $a\in A$ is $h(a')$, for some $a'\in A$, we have $\textbf{A}\vDash h(\varphi)(\textbf{h(a')})$ and so $\textbf{A}\vDash\varphi(\textbf{a'})$. Thus $a'\in S$ and $a\in h(S)$.
While the above seems to get the job done, I can't help shake the feeling I've done something illicit. Any help either confirming that I've got it or pointing out where I went wrong would be appreciated.