I need some help understanding part of my Model Theory notes:
"Given that $T$ is $\omega$-categorical and $\mathfrak{A} \vDash T$, for $S \subseteq A$, let $\langle S\rangle$ denote the smallest substructure of $\mathfrak{A}$ containing $S$. Suppose $S = \{ a_1, \ldots ,a_n\}$ and let $J_n$ be the collection of terms of $\mathcal{L}$ with variables amongst $v_1,\ldots,v_n$, one can easily show that $\langle S\rangle = \{ \tau^\mathfrak{A} [a_1, \ldots, a_n] : \tau \in J_n \}$."
I know that $\tau$ is a term and variables are terms, so $\tau^\mathfrak{A}[a_1,\ldots,a_n] = v_i^\mathfrak{A}[a_1,\ldots,a_n] = a_i$ shows that $S$ is contained within $\langle S\rangle$ as defined above. However, I do not know how to intuit $\langle S\rangle$ as above just from the information given. Also, I do not know how to show that $\langle S\rangle$ is of that form and is indeed a substructure.
Thanks in advance!