[[Material set theory]]
# Axiom Schema of Replacement

The **Axiom of Replacement**, technically an [[axiom schema]], is a possible axiom of [[Material set theory]] suggested by [[Abraham Fraenkel]] in the early 1920's[^2006]: #m/def/set/zf 
Let $\varphi(x,y)$ be a [[Class function]], i.e. a [[predicative formula]] such that $(\forall x)(\exists!y)\varphi(x,y)$.
Then,
$$
\begin{align*}
(\forall \shood A)(\exists \shood B)[y \in B \iff (\exists x \in A)\varphi(x,y)]
\end{align*}
$$
which is to say, the image of a set under a mapping is a set.

  [^2006]: 2006\. [[Sources/@moschovakisNotesSetTheory2006|Notes on set theory]], §11, p. 157


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