Maps between Classes, Continued: Agreement of Values

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $A$, $B$, $C$ and $D$ be classes. For a map $F$, $F(u)$ is its value at $u$, and $(u,v)$ is the ordered pair.

(i) Agreement of values. Let $F:A\to C$ and $G:B\to D$ be maps, and let $u$ be a class with $u\in A$ and $u\in B$. Then there is a set $v$ with $(u,v)\in F$ and $(u,v)\in G$ if and only if

$$F(u)=G(u).$$