Composite Maps into Classes, Images of Singletons, and Agreement of Maps by Induction

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $\mathbb{N}_{0}$, $0$, $1$ and the sum $n+1$ be as in The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion (ii) and The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion (iv). For a map $F$ between classes, $F(u)$ is its value at $u$; $(u,v)$ is the ordered pair, $A\times B$ the Cartesian product of classes $A$ and $B$, $R[D]$ the image of a class $D$ under a class $R$, $\{x\}$ the singleton of a set $x$, and $\subseteq$ the subclass relation.

(i) Let $X$ and $C$ be sets and $A$ and $B$ classes. For maps $W:A\times B\to C$, $f:X\to A$ and $g:X\to B$ there is a map $h:X\to C$ with

$$h(x)=W((f(x),g(x)))\quad\text{for every }x\in X.$$

(ii) For every class $R$ and all sets $x$ and $y$: $y\in R[\{x\}]$ if and only if $(x,y)\in R$.

(iii) Let $A$, $B$, $X$, $Y$ and $T$ be classes, and let $S:A\to X$ and $M:B\to Y$ be maps with $T\subseteq A$ and $T\subseteq B$. Suppose that $S(0)=M(0)$ holds whenever $0\in T$; and that, for every $k\in\mathbb{N}_{0}$, if $S(k)=M(k)$ holds whenever $k\in T$, then $S(k+1)=M(k+1)$ holds whenever $k+1\in T$. Then $S(k)=M(k)$ for every $k\in\mathbb{N}_{0}$ with $k\in T$.