[[Material set theory]]
# Domain Axiom for classes

The **Domain Axiom** is a possible axiom of [[Material set theory]] with [[Class|classes]][^2015]:
$$
\begin{align*}
(\forall \chood X)(\exists \chood Z)(\forall u)[u \in Z \iff (\exists v)[(u,v) \in X]]
\end{align*}
$$
which is to say, the domain of any [[Relation set]] $X$ exists,
unique by [[Axiom of Extensionality#Axiom of Extensionality for classes|extensionality]],
and denoted $\opn{dom} X$.

  [^2015]: 2015\. [[Sources/@mendelsonIntroductionMathematicalLogic2015|Introduction to Mathematical Logic]], §4.1, p. 235


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