Union of Bijections with Disjoint Domains and Codomains is Bijection/Corollary

Theorem
Let $A$, $B$, $C$, and $D$ be sets or classes.

Suppose that $A \cap B = C \cap D = \varnothing$.

Let $f: A \to C$ and $g: D \to B$ be bijections.

Then $f \cup g^{-1}: A \cup B \to C \cup D$ is also a bijection.

Proof
By Inverse of Bijection, $g^{-1}: B \to D$ is a bijection.

Thus the theorem holds by Union of Bijections with Disjoint Domains and Codomains is Bijection: