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:

$\Dom f = S$ and $\Cdm f = T$
$\Dom {f^{-1} } = T$ and $\Cdm {f^{-1} } = S$

That is:

$\Cdm {f^{-1} } = S = \Dom f$

$\blacksquare$