Proof Attempt:
For any random variable $X$ with non-decreasing continuous cdf $F(x)=\Pr(X≤x)$ (note that the inverse function does not necessarily exist due to flat regions in $F$), I wish to prove that the random variable $F(X)\sim U(0,1)$.
Note that because $F(x)$ is non-decreasing and is at least $0$ and at most $1$, $F(X)\le p \Longrightarrow X \le \max\{x:p=F(x)\}$ where $p∈[0,1]$.
Graphically, $(\max\{x:p=F(x)\},p)$ is simply the rightmost point of $F(x)$ at a specific magnitude $p$.
The cdf $G(p)$ of the random variable $F(X)$ is:
$$G(p)=\Pr(F(X)≤p)=\Pr(X≤\max\{x:p=F(x)\})=F(\max\{x:p=F(x)\})=p.$$
$G(p)=p:p∈[0,1]$ is the standard uniform cdf, so $F(X)\sim U(0,1)$.