In some part of a document I am writing, I first define a set
$$ S : A \to \mathcal P(B) \\ S(a) = \{ \text{complicated expression here using $a$} \} $$
Later, I prove that $\forall a,~\exists x, ~S(a) = \{ x \}$.
So, I want to now construct some notation for referring to the set with a single element $\{ x \} \subset B$. Something like
$$unset(S(a))$$
where $unset$ is some valid definition / notation.
On
It is just the union:
$$\bigcup S(a) = \bigcup\{x\} = x$$
Indeed, the union of a set $A$ is defined with $$c \in \bigcup A \iff \exists D \in A \text{ such that } c \in D$$
For two sets we have the more familiar notation $A \cup B = \bigcup \{A,B\}$.
There is no commonly used notation specifically for this, so you shouldn't hesitate to just make up your own. One way you can express it with standard notation is with the symbol $\bigcup$: if $S$ is a set, then $\bigcup S$ denotes the union of all the elements of $S$, so $\bigcup \{x\}=x$. This is kind of a "hack" though and is not something you can expect your readers to effortlessly understand. (Mathematically literate readers will figure it out, but it may take them a little work.)
Ultimately, the purpose of notation is to communicate, so you should pick your notation to communicate clearly and not be afraid to use words instead of notation if that would be clearer. I would probably recommend instead just writing something like:
There's nothing special about the choice of $s$ as the function name for this; it's a reasonable choice to use to remind the reader that it is related to $S$ but there's nothing wrong with using a different name if you have a better reason.