Axiom:Axiom of Replacement/Set Theory

Axiom
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:


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