TITLE: "Such that" symbol "$\mid$"
QUESTION [11 upvotes]: I know the "such that" symbol $\mid$ from the definition of sets:
$$\{x \mid x \in \Bbb N \land x < 3\}$$
Is it OK to use this symbol outside of sets. For instance, if I want to define a function that takes a non-empty set of natural numbers and yields the least element of this set, can I write:
$$f : \mathcal P (\Bbb N) \setminus \{ \emptyset\} \to \Bbb N \\
x \mapsto y \mid y \in x \land \forall z: z \in x \to z \geq y$$
Or would a mathematician shoot me on sight, if I wrote this?
EDIT:
Thank you for your comment. One proposition you made was to write "such that" in words. But doesn't this break the goal of a formal notation, i.e. its international comprehension. If I wrote:
$$x \mapsto y \text{ tal que } y \in x \land \forall z: z \in x \to z \geq y$$
or
$$x \mapsto y \text{ tal que } y \text{ sea el elemento mínimo del conjunto } x$$
Wouldn't this lead to misunderstandings if the reader didn't speak Spanish?
To make the question short: How would you write down the function $f$ as defined above?

REPLY [2 votes]: At the university I attended, s.t. and .э. (some profs omitted the periods on the later case) were used. However, I doubt these are universal and will mimic others when saying written language is usually the best way to convey meaning. Coming from a statistics background, for example, | typically means "given" to me rather than "such that" like some people use in set notation (I use a colon as "such that" in set notation).
Also, things like э can be used as, or at least look familiar to, "belongs to" notations for sets. Mathematicians generally use, recycle, and then reuse notations again and again, so in a situation where you're speaking to a general audience-- words are your best friend.