Showing that an intersection of indexed sets is a subset of every individual indexed set

Question

I am required to show that for every $k \in I$, $\bigcap_{i\in I}A_{i}\subseteq A_{k}$ where $I$ is an index for a collection of subsets $A_{i}\subseteq S$, $i \in I$.
This seems obvious to me from the definition of intersection. That is:
$\bigcap_{i\in I}A_{i}= \{x \in S: \forall i \in I, x \in A_i\}$.
Since every element in the set is, by definition, in $A_i$ for all $i \in I$, it is, of course, also in $A_k, k \in I$. Therefore, the set is a subset of $A_k$.
Is this all I need to show? Or is there an actual proof I can/should write?