# Category:Axioms/Axiom of Replacement

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This category contains axioms related to Axiom of Replacement.

### 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.

## Pages in category "Axioms/Axiom of Replacement"

The following 11 pages are in this category, out of 11 total.

### R

- Axiom:Axiom of Replacement
- Axiom:Axiom of Replacement (Class Theory)
- Axiom:Axiom of Replacement (Set Theory)
- Axiom:Axiom of Replacement/Also known as
- Axiom:Axiom of Replacement/Also presented as
- Axiom:Axiom of Replacement/Class Theory
- Axiom:Axiom of Replacement/Class Theory/Formulation 1
- Axiom:Axiom of Replacement/Class Theory/Formulation 2
- Axiom:Axiom of Replacement/Class Theory/Formulation 3
- Axiom:Axiom of Replacement/Set Theory