Domain of Bijection is Codomain of Inverse

Theorem
Let $S$ and $T$ be sets.

Let $f: S \to T$ be a bijection.

Let $f^{-1}: T \to S$ be the inverse of $f$.

Then the codomain of $f^{-1}$ equals the domain of $f$.

Proof
Follows directly from the definition of domain and codomain:


 * $\operatorname{Dom} \left({f}\right) = S$ and $\operatorname{Cdm} \left({f}\right)= T$
 * $\operatorname{Dom} \left({f^{-1}}\right) = T$ and $\operatorname{Cdm} \left({f^{-1}}\right)= S$

That is:
 * $\operatorname{Cdm} \left({f^{-1}}\right) = S = \operatorname{Dom} \left({f}\right)$

Also see

 * Codomain of Bijection is Domain of Inverse