User:Dfeuer/Axiom of Replacement/Gödel

Axiom
$\forall x,A: \left({ \mathfrak{Un}(A) \implies \exists y: \forall v: \left({v \in y \iff \exists u: (u \in x \land (u, v) \in A) }\right) }\right)$

where $\mathfrak{Un}(A) \iff \forall u, v, w: \left({ (u, v) \in A \land (u, w) \in A \implies v = w }\right)$.

That is, for each set $x$ and each many-to-one relation $A$, there exists a set $y$ such that for all $v$, $v \in y$ iff there exists a $u \in x$ such that $(u, v) \in A$.

Remarks
Note that $A$ need not be a mapping on $x$. No assumption whatsoever is made about the domain of $A$.

Bernays and Gödel define a many-to-one relation inversely to. For the sake of consistency with the rest of the site, their definition of $\mathfrak{Un}$ and the statement of the axiom have been reversed.