Mathematical symbols for common phrases such as $\textit{for some}$ and $\textit{such that}$

Question

I noticed that I use the phrases $\textit{for some}$ and $\textit{such that}$ a lot in pretty much every proof.

Here is an example of the use of $\textit{for some}$:

$\textit{The range of a function is defined as:}$

$$f(X) = \left \{ y \in Y \ | \ y = f(x) \ \text{for some} \ x \in X \right \}.$$

Are there common mathematical symbols for those phrases similar to $\exists \ \text{and} \ \forall$?

Answer 1

You can read $\{y\in Y\mid \exists x\in X, y=f (x)\}$ as "the set of all $y\in Y$ such that $y=f (x)$ for some $x\in X$"; essentially, $\mid$ can function as "such that" in that context.

A simple "s. t. " is often understood as "such that" too.

Answer 2

If I needed or wanted to use terse symbols, I would render $\text{“} y = f(x)$ for some $x\in X\text{''}$ as $$ \exists x\in X\ y = f(x). $$

(A typographical note: In MathJax and LaTeX in expressions like $\forall\varepsilon\, \exists\delta\,[\cdots\cdots],$ I manually add a bit of space after $\forall\varepsilon$ and after $\exists\delta$ for reasons that I expect will be clear by hindisight, although I sometimes see this neglected by those who use MathJax and LaTeX.)