I have a question regarding how to precisely define a formula of mine.
I have a function which maps words to positive integers, $f: S \rightarrow \mathbb{Z}^+$. When I apply this mapping funtion to all elements of a set $ \mathcal{D} = \{s_i\}$, where $s_i$ are all words in a fixed sized vocabulary. Then this mapping function yields a set of positive, unique integers for all of the elements in $\mathcal{D}$. Would this notation be correct: $s_i \in \mathcal{\{D\ | \exists! s_i f(s_i) \}}$
So from the comments, my understanding is that $\mathcal D$ is a subset of $S$ which is the set of all words, i.e. $\mathcal D \subset S$.
To answer your question, there are two correct notations for the same set you are asking about, namely: $$f(\mathcal D) = \{f(s_i) \mid s_i \in \mathcal D\} $$ The left side of the equation, in words, is "the image under $f$ of the subset $\mathcal D$". The right hand side, in words, is "The set of all $f(s_i)$, such that $s_i$ is an element of $\mathcal D$."
The general pattern here applies to any function $f : X \to Y$, and any subset $A \subset X$: $$f(A) = \{f(x) \mid x \in A\} $$ which automatically yields a subset of $Y$.