Inverse Element of Bijection

Theorem
If $f: S \to T$ is a bijection, then $f^{-1} \left({y}\right) = x \iff f \left({x}\right) = y$.

Proof
Suppose $f$ is a bijection.

Because $f^{-1}$ is a bijection from Bijection iff Inverse is Bijection, it is by definition a mapping, and the result follows directly from Inverse Mapping Image.