Sequences: Enlarging the Codomain, Extensionality and the Range

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let $A$, $B$ and $X$ be classes, not necessarily sets. For every map $c:\mathbb{N}\to X$ and $n\in\mathbb{N}$, write $c_{n}=c(n)$ for the value of $c$ at $n$. $\subseteq$ is the subclass relation and $\operatorname{ran}a$ the range of $a$.

(i) Enlarging the codomain. If $A\subseteq B$, then every map $a:\mathbb{N}\to A$ is a map $a:\mathbb{N}\to B$. In particular, for sets $A\subseteq B$, every sequence in $A$ is a sequence in $B$.

(ii) Extensionality. Let $a,b:\mathbb{N}\to X$ be maps with $b_{k}=a_{k}$ for every $k\in\mathbb{N}$. Then $b=a$. In particular, when $X$ is a set, two sequences in $X$ with the same terms are equal.

(iii) Range. Let $a:\mathbb{N}\to X$ be a map. For every class $y$,

$$y\in\operatorname{ran}a\iff\text{there is }n\in\mathbb{N}\text{ with }y=a_{n}.$$