Membership in Inverse Relations, Preimages and Images of Classes, and Enlarging the Codomain of a Map

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let $R$, $C$, $F$, $A$, $B$ and $D$ be classes, not necessarily sets. $R^{-1}$ is the inverse of $R$, $F[A]$ is the image of $A$ under $F$, and $R^{-1}[C]$ is the preimage of $C$ under $R$.

(i) For all sets $x$ and $y$: $(x,y)\in R^{-1}$ if and only if $(y,x)\in R$.

(ii) For every set $x$: $x\in R^{-1}[C]$ if and only if there is a set $w$ with $w\in C$ and $(x,w)\in R$.

(iii) If $F:A\to B$ is a map, then for every set $x$: $x\in F^{-1}[C]$ if and only if $x\in A$ and $F(x)\in C$, where $F(x)$ is the value of $F$ at $x$.

(iv) If $F:A\to B$ is a map, then $F[A]\subseteq B$.

(v) If $F:A\to B$ is a map and $F(u)\in D$ for every $u\in A$, then $F:A\to D$ is a map.