Axiom of replacement

Axiom schema of replacement

The axiom schema of replacement asserts that the image of a set under any definable function will also fall inside a set.

Formally, let \phi be any formula in the language of ZFC whose free variables are among {\displaystyle x,y,A,w{1},\dotsc ,w{n)), so that in particular B is not free in \phi . Then:

{\displaystyle \forall A\forall w{1}\forall w{2}\ldots \forall w_{n}{\bigl [}\forall x(x\in A\Rightarrow \exists !y\,\phi )\Rightarrow \exists B \forall x{\bigl (}x\in A\Rightarrow \exists y(y\in B\land \phi ){\bigr )}{\bigr ]}.} In other words, if the relation \phi represents a definable function f, A represents its domain, and f(x) is a set for every x\in A, then the range of f is a subset of some set B. The form stated here, in which B may be larger than strictly necessary, is sometimes called the axiom schema of collection.