nLab range

Classically, the range of a function $f$ with domain $A$ is the set $\{f(x) \;|\; x \in A\}$ (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
  $$\{y \colon B \;|\; \exists x\colon A,\; y = f(x)\}$$

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

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.