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Simulation of a CDF from a uniform distribution of its values

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I need a mathematical proof for why pdf is generated from an inverse of cdf approach. I only want a mathematical proof and if it is an approximation please a proof for why the approximation holds. I specifically need to emphasize the part of the proof for why this manipulation of F gives values that is along the pdf, f, of cdf, F because this is my main problem.

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This solution explains in small detail the relation between the cumulative probability distribution of a random variable and the uniformity of the distribution of of the values of the CDF.

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