range in nLab

Classically, the range of a function ff with domain AA is the set {f(x)|x∈A}\{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:B|∃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→Bf\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.