# Axiom:Axiom of Replacement/Class Theory

< Axiom:Axiom of Replacement(Redirected from 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.

Symbolically:

- $\forall Y: \map {\text{Fnc}} Y \implies \forall x: \exists y: \forall u: u \in y \iff \exists v: \tuple {v, u} \in Y \land v \in x$

where:

- $\map {\text{Fnc}} X := \forall x, y, z: \tuple {x, y} \in X \land \tuple {x, z} \in X \implies y = z$

and the notation $\tuple {\cdot, \cdot}$ is understood to represent Kuratowski's formalization of ordered pairs.

### 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**.