Ordered Pairs in a Map between Classes: Membership as a Value, and the Components Lie in the Domain and the Codomain

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let $A$, $B$ and $F$ be classes, with $F$ a map $F:A\to B$. Values $F(a)$ are as in Functions, Values of a Function, and Functions from One Class to Another (ii), and $(a,b)$ is the ordered pair.

(i) For every $a\in A$ and every set $b$,

$$(a,b)\in F\iff F(a)=b.$$

(ii) For all sets $a$ and $b$, if $(a,b)\in F$, then $a\in A$ and $b\in B$.