This can be easily proved using the probability integral transform theorem:

Suppose that Y is a continuous random variable with pdf f(y) and continuous cdf F(y). Suppose that X is a continuous standard uniform random variable. Then is a random variable with the same probability distribution as Y. (For proof see for example Roussas G. G. (1997) A Course in Mathematical Statistics. Or just use a search engine).

In your case, let Y be an exponential random variable with pdf . Note: You will need to realise that If X ~ U(0, 1) then 1 - X ~ U(0, 1).