Inverse Element of Bijection

Theorem

Let $S$ and $T$ be sets.

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

Then:

$\map {f^{-1} } y = x \iff \map f x = y$

where $f^{-1}$ is the inverse mapping of $f$.

Proof

Suppose $f$ is a bijection.

Because $f^{-1}$ is a bijection from Bijection iff Inverse is Bijection, it is by definition a mapping.

The result follows directly from Image of Element under Inverse Mapping.

$\blacksquare$