"Such that" symbol "$\mid$"

You ask how I would write this function:

$$f : \mathcal P (\Bbb N) \setminus \{\} \to \Bbb N \\ x \mapsto y \mid y \in x \land \forall z: z \in x \to z \geq y$$

First I’d correct the error in the top line: you want the domain to be the family of non-empty subsets of $\Bbb N$, which is $\wp(\Bbb N)\setminus\{\varnothing\}$ or, if you insist on avoiding the standard notation for the empty set, $\wp(\Bbb N)\setminus\{\{\}\}$. Your $\wp(\Bbb N)\setminus\{\}=\wp(\Bbb N)\setminus\varnothing=\wp(\Bbb N)$. The rest is easily compressed into one line:

$$f:\wp(\Bbb N)\setminus\{\varnothing\}\to\Bbb N:x\mapsto\min x\;.$$

In my view $y=\min x$ is much easier to grasp than ‘$y$ is the unique element of $x$ such that $y\le z$ for all $z\in x$’, whether the latter is expressed in English, in Spanish, or entirely in mathematical symbols.

For the more general question, I would no more use $\mid$ for such that in general than I would use the colon that I prefer for my set notation: I would not expect it to be automatically understood (and would not immediately understand it myself). In the given context I would understand tal que immediately, and my Spanish is very, very minimal.

I don’t think consider international comprehensibility to be a major goal of mathematical notation, formal or (relatively) informal. The primary function of good mathematical notation in everyday mathematical use is to make the mathematics easier to understand and follow. (Notation intended to aid mechanical theorem-proving or the like is an exception.)

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 $\min$.