If $T$ is surjective, then Theorem RSLT says $\rng{T}=V$, so $\rank{T}=\dimension{V}$ by Definition ROLT. In turn, the hypothesis gives $\rank{T}=\dimension{U}$. Then, using Theorem RPNDD,
\begin{equation*}
\nullity{T}=\left(\rank{T}+\nullity{T}\right)-\rank{T}=\dimension{U}-\dimension{U}=0
\end{equation*}
With a null space of zero dimension, $\krn{T}=\set{\zerovector}$, and by Theorem KILT we see that $T$ is injective. $T$ is both injective and surjective so by Theorem ILTIS, $T$ is invertible.