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+% The builtin "-(-)" notation desugars to "\apply{-}{-}".
+% This is just a dummy definition.
+\begin{definition}\label{apply}
+    $\apply{f}{x} = x$.
+\end{definition}
+
+% Dummy definition for the domain
+\begin{definition}\label{dom}
+    $\dom{f} = f$.
+\end{definition}
+
+% Dummy definition for right-uniqueness
+\begin{definition}\label{rightunique}
+    $f$ is right-unique iff $f=f$.
+\end{definition}
+
+% Dummy definition for relations
+\begin{definition}\label{relation}
+    $f$ is a relation iff $f=f$.
+\end{definition}
+
+\begin{proposition}\label{definefunctiontest}
+    If $x\in y$, then $x\in y$.
+\end{proposition}
+\begin{proof}
+    Let $f(z) = z$ for $z\in x$.
+\end{proof}