Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number such that

(i)	$F : \mathbb{R} \to [0, 1]$ is a D205: Probability distribution function for $X$
(ii)	$F$ is an D3393: Invertible function with an D4024: Inverse function $F^{-1} : [0, 1] \to \mathbb{R}$

Let $U \overset{d}{=} \text{Uniform}[0, 1]$ be a D4624: Standard unsigned uniform random real number.

If $x \in \mathbb{R}$, then \begin{equation} \begin{split} \mathbb{P}(F^{-1}(U) \leq x) & = \mathbb{P}(F F^{-1}(U) \leq F(x)) \\ & = \mathbb{P}(U \leq F(x)) \\ & = F_U(F(x)) \\ & = F(x) \end{split} \end{equation} where $F_U$ is a distribution function for $U$. $\square$