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.