Inverse Element of Bijection

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

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

Then:
 * $f^{-1} \left({y}\right) = x \iff f \left({x}\right) = y$

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.