If the domain of a function is any set, how is it formally defined?

Question

If the domain of a function $f$ is any set (for example, cardinality operator), then $f$ is defined as $A\to B$ where $A$ is the set of all sets, but that set does not exist, so how is it formally defined? Is $f:=S\mathop{\mapsto} f(S)$ a valid definition?

Answer 1

I assume that you're working in the ZFC axioms of set theory or something similar. Note that there are set theories in which the class of all sets is in fact a set, such as Quine's New Foundations.

A function is, by definition, a set of ordered pairs. So a function can't have a proper class as its domain. So the "function" $X\mapsto\vert X\vert$ is not in fact a function at all. Instead, it's a class function - a class of ordered pairs such that [stuff].

Answer 2

If $f(A)$ is the cardinality of $A$ or the power set or whatever, then officially $f$ is not a function, it is a "class function" (meaning roughly "a function, except its domain is a proper class, not a set". (For example the class of all sets is a "proper class")).

Answer 3

The cardinality operator is not a function.

It connects every set with some object.

Likewise $\wp$ and $\cup$ can be looked at as operators (not functions) on sets.