TITLE: What is meant by "function" in the axiom schema of replacement.
QUESTION [11 upvotes]: Axiom schema of replacement: Let the domain of the function $F$ be the set $A$. Then the range of $F$ (the values of $F(x)$ for all members $x$ of $A$) is also a set. — Tarski–Grothendieck set theory

I can't fully appreciated the axiom; for me, when talking about the function $F$, we should assume a domain and a range before.  So sounds that the axiom is circular.
So my question is: what is a function?
If a function is defined not through sets, then it sounds that “function” is more fundamental concept than “set.”

REPLY [3 votes]: If the quoted axiom were part of a theory of sets and classes, like Morse-Kelley or Neumann-Bernays-Gödel, then I'd simply say that a function is a class of ordered pairs in which no two distinct ordered pairs have the same first component.  But in such a theory, this form of replacement would be a single axiom, not an axiom schema.  Since it's labeled as a schema in the question, I must assume that we're dealing with a theory of sets, not classes, like Zermelo-Fraenkel set theory.  In that context, this usage of the word "function" is rather unusual but not unheard-of.  I'd say it's unusual enough that any text using it should give an explanation or at least a pointer to an explanation.  (Suggestion: Look for such an explanation in the source where you found this formulation of the replacement schema.)  The usual meaning of "function" in ZF (or similar theories) is a set of ordered pairs, no two of which have the same first component, and that is definitely not what is meant here.  As some of the previous answers have explained, the intended notion of "function" here is what I'd call a parametrically definable operation.  Technically, this consists of two things:  First, there is a (first-order) formula $\phi(x,y,\vec p)$ with two "important" free variables $x$ and $y$ and possibly some additional "auxiliary" variables or "parameters" $\vec p$; second there are specific sets as values for those parameters $\vec p$.  To say that the formula $\phi$ with those values of $\vec p$ constitutes a "function" is to say that the formula 
$$
\forall x\,\forall y\,\forall z\,[(\phi(x,y,\vec p)\land\phi(x,z,\vec p))\implies y=z]
$$
is true (for the given values of $\vec p$).  This mixture of syntax (\phi) and semantics (values for $\vec p$) is rather awkward to formalize carefully (though it has been done), so, when introducing the axioms of set theory, I prefer not to introduce these strange "functions" and instead to formulate replacement as Lord_Farin did.