Axiom:Axiom of Replacement/Set Theory

Axiom
For any function $f$ and subset $S$ of the domain of $f$, there is a set containing the image $\map f S$.

More formally, let us express this as follows:

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

That is, we have:
 * $\forall y: \exists x: \forall z: \paren {\map P {y, z} \iff x = z}$.

Then we state as an axiom:


 * $\forall w: \exists x: \forall y: \paren {y \in w \implies \paren {\forall z: \paren {\map P {y, z} \implies z \in x} } }$

Also presented as
The two above statements may be combined into a single (somewhat unwieldy) expression:


 * $\paren {\forall y: \exists x: \forall z: \paren {\map P {y, z} \implies x = z} } \implies \forall w: \exists x: \forall y: \paren {y \in w \implies \forall z: \paren {\map P {y, z} \implies z \in x} }$

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