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\begin{signature}\label{phase5_local_pair}
$(x,y)$ is a set.
\end{signature}
\begin{proposition}\label{phase5_local_function}
For all $A$ we have $A = A$.
\end{proposition}
\begin{proof}
Fix $A$.
Let $f(x) = x$ for $x\in A$.
Show for all $y$ we have if $y\in A$, then $(y,y)\in f$.
\begin{subproof}
Fix $y$.
Assume $y\in A$.
Follows by assumption.
\end{subproof}
Follows by assumption.
\end{proof}
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