Given $f$ is invertible. We need to show that $f$ has a unique inverse.

If a function $f$ has two inverses $g_1$ and $,g_2 $ then $fog_1 (y)=I_y\;and\;fog_2(y)=I_y$ for $y \in Y$

$\Rightarrow fog_1 (y)=I_y(y)=fog_2(y)$

$\Rightarrow f(g_1(y))=f(g_2(y))$

Since $f $ is invertible $f$ is one-one, $\Rightarrow g_1(y)=g_2(y)$

There if $f$ has two inverses, then $g_1=g_2$. In otherwords, $f$ will only have a unique inverse.