probability - Distribution of continuous distribution function of random variable

Suppose we have $X$ - random variable with distribution function $F(x)$, where $F(x)$ -- continuous distribution function. How to find the distribution function of new random variable $Y = F(X)$?

Important : There is no assumptions that $F(x)$ is strict monotone. So we can't just find inverse function $F^{-1}(x)$.

P.S. I know the answer: $U_{[0;1]}$, but how to prove it? I can prove it only in case, when $F(x)$ - strict monotone.

Answer

Let $u\in\left(0,1\right)$.

If $z_{u}:=\inf\left\{ x\in\mathbb{R}\mid F\left(x\right)\geq u\right\} $
then $F\left(z_{u}\right)\geq u$ since $F$ is right-continuous.

So we have $F(x)

If moreover $F$ is continuous then $P\left(X

Then $P\left(F\left(X\right)