If $\displaystyle f: A \to B $ is bijective show that $\displaystyle f^{-1} $ is unique.

Suppose that $\displaystyle f^{-1} $ and $\displaystyle h $ are inverses of $\displaystyle f $. Let $\displaystyle y = f(x) $. We need to show that $\displaystyle f^{-1}(y) = h(y) $ for all $\displaystyle y \in B $. So $\displaystyle f^{-1}(f(x)) = h(f(x)) = x $ for all $\displaystyle y \in B $, thus $\displaystyle f^{-1} $ is unique?