Codomain of Bijection is Domain of Inverse

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

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

$\blacksquare$


Also see


Sources