"Such that" symbol "$\mid$" 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?

I would go a bit farther even than Brian's answer, and use more words and fewer symbols. (I'm not sure that his second colon is standard, by the way.)
I would simply say "given a nonempty set $A$ of natural numbers, we denote its least element by $f(A)$."  As Brian says, this particular function $f$ is often simply called $\min$.
