Relations, Domain, Range, Inverse and Composition

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let $R$, $S$ and $A$ be classes. The classes below are formed by class abstraction from formulas that quantify over set variables only, with the ordered pair $(u,v)$ used as a defined set symbol; these formulas are therefore predicative, as Class Theory NBG: the Axioms, Standing Conventions and Basic Notation §comprehension requires.

¶ $R$ is a relation if every element of $R$ is an ordered pair, that is, for every $p\in R$ there are sets $u$ and $v$ with $p=(u,v)$. For sets $u$ and $v$, $u\mathrel{R}v$ means $(u,v)\in R$.

¶ The domain of $R$ is the class

$$\operatorname{dom}R=\{u:\exists v\,((u,v)\in R)\}.$$

¶ The range of $R$ is the class

$$\operatorname{ran}R=\{v:\exists u\,((u,v)\in R)\}.$$

¶ The inverse of $R$ is the class

$$R^{-1}=\{p:\exists u\,\exists v\,((u,v)\in R\wedge p=(v,u))\}.$$

¶ The composition of $R$ followed by $S$ is the class

$$S\circ R=\{p:\exists u\,\exists v\,\exists w\,((u,v)\in R\wedge(v,w)\in S\wedge p=(u,w))\}.$$

¶ A relation on $A$ is a relation $R$ with $R\subseteq A\times A$, where $\subseteq$ is the subclass relation and $A\times A$ is the Cartesian product.