Gluing Maps on Disjoint Domains, and Shrinking the Codomain

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $A$ and $C$ be sets and $X$ a class, with maps and their values as in Functions, Values of a Function, and Functions from One Class to Another, and the image $f(A)$ of $A$ under a map $f$.

(i) Gluing. Let $f:A\to X$ and $g:C\to X$ be maps with $A\cap C=\emptyset$. Then $f\cup g$ is a map $f\cup g:A\cup C\to X$, with $(f\cup g)(x)=f(x)$ for $x\in A$ and $(f\cup g)(x)=g(x)$ for $x\in C$.

(ii) Gluing bijections. Let $B$ and $D$ be sets with $B\cap D=\emptyset$, and let $f:A\to B$ and $g:C\to D$ be bijections with $A\cap C=\emptyset$. Then $f\cup g$ is a bijection $f\cup g:A\cup C\to B\cup D$.

(iii) Shrinking the codomain. Let $f:A\to X$ be a map and $Y$ a class with $f(A)\subseteq Y$. Then $f:A\to Y$ is a map. If $f$ is injective, then $f:A\to f(A)$ is a bijection.