This idea is formalized in the Glivenko-Cantelli theorem (also called the fundamental theorem of statistics), which states that the EDF converges pointwise to the parent distribution.

This result is illustrated here for three different parent distributions: the normal distribution with parameters (0, 1), the uniform distribution with parameters (-5, 5), and the gamma distribution with parameters (2, 0.6). You can see the EDF of a random sample approach the parent distribution as the sample size increases.