i am currently reading about basic set theory, especially the Zermelo–Fraenkel set theory, peano axioms and the Von Neumann ordinals. In my reference book (Einführung in die Mengenlehre by H.-D. Ebbinghaus, its in german) functions are introduced as mappings, essentially an input-output tuple. I am currently halfway into the book and i am wondering how well-defined the concepts introduced are. Since there are some problems with unrestricted set-comprehension (Russels antiome) and as a consequence a lot of work went into creating ZFC, i was wondering what this means for functions.
For basic arithmetic functions i can create the equivalent inductive set by easily by resusing the definition of Von Neuman ordinals Successor function and modifying it a bit (if we accept this as the current peano-structure is the set-equivalent of f(x) = x + 1).
I am think we are somewhat limited since i can't encode f(y) = {x | (x not element of x) and (x is not element of y)} (its not valid in ZFC), but it's not really useful and i struggle to find more useful examples.
I claim that any mapping whose domain is a set, and which outputs sets, can be treated in ZFC.
First, let's look at what happens with the $f$ you propose. The problem is that, for any $y$, the "set" $\{x: x\not\in x, x\not\in y\}$ is not, in fact, a set! (Presumably this is what you mean when you say this is "not valid in ZFC.) So it's not the sort of thing that ZFC is trying to talk about (but see below). If we tweak it so that it always outputs sets, e.g., $$g(y)=\{x: x\in y, x\not\in x\},$$ then we run into the second problem: its domain is $\{$all sets$\}$, which is not a set! So we need to restrict it further: for any set $z$, the set $g_z$ with domain $z$, given by $g_z(y)=\{x: x\in y, x\not\in x\}$ for $y\in z$, is indeed something ZFC can code: $g_z$ is identified with the set $$\{(y, y): y\in z\}$$ (since in ZFC, no set contains itself, we have $g_z(y)=y$ for all $y\in z$).
The key "function-forming" axiom is the axiom scheme of Replacement. Replacement says (roughly) that whenever I have some rule for assigning a unique $y_x$ to every $x\in z$ for some domain $z$, then the function $\{(x, y_x): x\in z\}$ exists. (This isn't how Replacement is usually phrased, but they're equivalent.) So: any rule you can define for taking in sets and spitting out sets, with domain a set, corresponds to a function in ZFC.
Now, what about "functions" defined on all sets? E.g. the $g$ I defined above, or the cardinality map $x\mapsto \vert x\vert$. These are not functions, but rather class functions, and while not as nice as functions, they can still be worked with in ZFC; see e.g. this question. We can even work with functions like your $f$, which output classes instead of sets (we treat these as total class relations). ZFC is much more robust than you give it credit for; if you go through the early chapters of Kunen's book or Jech's book, for example, you'll get a good sense of this.