# Axiom:Axiom of Replacement/Class Theory

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## Axiom

### Formulation 1

For every mapping $f$ and set $x$ in the domain of $f$, the image $f \sqbrk x$ is a set.

### Formulation 2

For every mapping $f$ and set $x$, the restriction $f \restriction x$ is a set.

### Formulation 3

Every mapping whose domain is a set is also a set.

## Also known as

The **axiom of replacement** is also known as the **axiom of substitution**.

## Also see

- Results about
**the Axiom of Replacement**can be found**here**.

## Historical Note

The **axiom of replacement** was added to the axioms of **Zermelo set theory** by Abraham Halevi Fraenkel, and also independently by Thoralf Albert Skolem.

The resulting system of axiomatic set theory is now referred to as **Zermelo-Fraenkel Set Theory**.