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.