nLab
range

Classically, the range of a function $f$ with domain $A$ is the set $\{f(x)\mid x\in A\}$ (whose existence, in material set theory, is given by the axiom of replacement). As we came to realise that a function should be given with a codomain (which is automatic in structural set theory), the term ‘range’ generalised in two ways:

• as the codomain itself, so that the earlier terminology is then preserved only for surjections;

• as the image

  $$\{y:B\mid \exists x:A,y=f(x)\}$$\{y \colon B \;|\; \exists x\colon A,\; y = f(x)\}

  (whose existence, in axiomatic set theory, is given by the much weaker axiom of bounded separation) of $f:A\to B$.

The former generalisation was historically common (and is sometimes still used) in groupoid theory; the latter is what we usually mean today.

Note that the axiom of replacement is still needed for a function (such as a family of sets) whose codomain is a proper class, to prove that its image is small when its domain is small.