This technique is called inverse transform sampling. More details: https://en.wikipedia.org/wiki/Inverse_transform_sampling#Intuitions.
The trick behind the proof is the probability of X<=x is equal to the probability of T(U) <= x, which is equal to the probability of U <= T_inverse(x). ( let X = T(U), where U is uniform distribution and T is our desired transform ([0,1] -> R).
This technique is called inverse transform sampling. More details: https://en.wikipedia.org/wiki/Inverse_transform_sampling#Intuitions. The trick behind the proof is the probability of X<=x is equal to the probability of T(U) <= x, which is equal to the probability of U <= T_inverse(x). ( let X = T(U), where U is uniform distribution and T is our desired transform ([0,1] -> R).