Probability – Why does the value of the CDF in the corresponding range remain the same even after the application of a non-linear transformation?

For random variable transformations, why the value of the CDF in the corresponding range remains the same even after the application of a nonlinear transformation?

For example. X ~ U (0, 2), the PDF of X is $ f_X (x) = frac {1} {2} $ and the CDF is $ F_X (x) = frac {x} {2} $. Let Y = $ X ^ 2 $, I can finish derivative $ f_Y $ and $ F_y $ by myself. However, I do not understand why $ F_X (x) = F_Y (x ^ 2) $ is always true.