# Axiom:Axiom of Replacement

## Axiom

### Set Theory

For every mapping $f$ and subset $S$ of the domain of $f$, there exists a set containing the image $f \sqbrk S$.

More formally, let us express this as follows:

Let $\map P {x, z}$ be a propositional function, which determines a mapping.

That is, we have:

- $\forall x: \exists ! y : \map P {x, y}$.

Then we state as an axiom:

- $\forall A: \exists B: \forall y: \paren {y \in B \iff \exists x \in A : \map P {x, y} }$

### Class Theory

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.

## 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**.

## Sources

- 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 9$ Zermelo set theory