Axiom:Axiom of Replacement/Class Theory

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

• Equivalence of Formulations of Axiom of Replacement

• 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.