Editing Zermelo–Fraenkel set theory (section)

=== Axiom schema of replacement ===
{{Main|Axiom schema of replacement}}

The axiom schema of replacement asserts that the [[image (mathematics)|image]] of a set under any definable [[Function (mathematics)|function]] will also fall inside a set.

Formally, let <math>\varphi</math> be any [[Well-formed formula|formula]] in the language of ZFC whose [[free variable]]s are among <math>x, y, A, w_1, \dotsc, w_n,</math> so that in particular <math>B</math> is not free in <math>\varphi</math>. Then:

<div style="margin-left:1.6em;"><math>\forall A\forall w_1 \forall w_2\ldots \forall w_n \bigl[\forall x ( x\in A \Rightarrow \exists! y\,\varphi ) \Rightarrow \exists B \ \forall x \bigl(x\in A \Rightarrow \exists y (y\in B \land \varphi)\bigr)\bigr].</math></div>

(The [[Uniqueness quantification|unique existential quantifier]] <math>\exists!</math> denotes the existence of exactly one element such that it follows a given statement.)

In other words, if the relation <math>\varphi</math> represents a definable function <math>f</math>, <math>A</math> represents its [[domain of a function|domain]], and <math>f(x)</math> is a set for every <math>x \in A,</math> then the [[range of a function|range]] of <math>f</math> is a subset of some set <math>B</math>. The form stated here, in which <math>B</math> may be larger than strictly necessary, is sometimes called the [[Axiom schema of replacement#Axiom schema of collection|axiom schema of collection]].