The Image and the Preimage of a Class under a Class

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let $R$, $F$, $A$ and $B$ be classes, and let $(u,v)$ denote the ordered pair.

¶ The image of $A$ under $R$ is the class

$$R[A]=\{v:\exists u\,(u\in A\wedge(u,v)\in R)\},$$

formed by class abstraction; the formula quantifies over set variables only, with $(u,v)$ a defined set symbol, so it is predicative as Class Theory NBG: the Axioms, Standing Conventions and Basic Notation §comprehension requires.

¶ The preimage of $B$ under $F$ is the image of $B$ under the inverse $F^{-1}$ of $F$, that is, the class $(F^{-1})[B]$ given by the image clause above with $R$ taken to be the class $F^{-1}$; it is written $F^{-1}[B]$.