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.)
This notion of "replacing" an element in a set $S$, while maintaining the same name $S$ for the result, is an example of what assignment statements do in imperative programming languages. You're thinking of $S$ as a variable in such a language, where at any point in time $S$ has a state, and operations mutate that state.
Variables in mathematics typically aren't thought of or used in that way (nor are variables in functional programming languages). The result of removing $a$ from $S$ and adding $b$ to that is another set, call it $S'$:
$$
S' = (S\setminus\{a\}) \cup \{b\}.\tag{*}
$$
The righthand side of (*) is the operation on $S$ that you describe.
If you wanted to be imperative about it and redefine $S$, I suppose you might change "$S' = $" in (*) to "$S := $" or "$S\leftarrow $". You can do that in a description of an algorithm, but I'd say don't do it in a proof — "it's just not done", and you'll confuse people.
In model theory, there's a related notion, and a notation for it. An assignment is a function $\mathsf{a}\colon\mathsf{Variables}\to U$ from variables to values in a set $U$. Two common scenarios: in propositional logic, $\mathsf{Variables}$ is the set of propositional variables, and $U$ is the set of truth values; in first order logic, $\mathsf{Variables}$ is the set of variables for individuals and $U$ is the universe of a model.
If $\mathsf{a}$ is an assignment, $v$ a variable, and $u\in U$, the notation $\mathsf{a}[v\backslash u]$ is used to denote the assignment $\mathsf{a}'$ which agrees with $\mathsf{a}$ on every variable except possibly $v$, and such that $\mathsf{a}'(v) = b$. This is just replacement of an element of $\mathsf{a}$ by another: $(v, \mathsf{a}(v))$ is replaced by $(v,b)$. So, yes it's similar — but not exactly the same as what you describe, as this notation applies only to functions. In any case, the result of the replacement is rightly regarded as a different assignment of values to variables; the name $\mathsf{a}$ doesn't all of a sudden refer to this different thing.
Best Answer
"Such that" is occasionally denoted by
\ni= $\,\ni\,$, e.g., in lecture, to save time, as a shortcut. Others, when writing in lectures or taking notes, and again, to save time, use "s.t.".But in writing anything to submit (homework, publication), when possible, it is best to just write the words "such that".
In sets though, like set-builder notation, both $\mid$ and $:$ are used:
$$\{x \in \mathbb R \mid x < 0\}$$ $$\{x \in \mathbb R : x \lt 0\}$$
"The set of all $x \in \mathbb R$ such that $x \lt 0$.