The two Hohenberg-Kohn theorems are the cornerstone of DFT. Let us examine the first theorem:
Go to the Density Functional Theory entry in Wikipedia or another encyclopedia of your choice, and read what is written there about the first Hohenberg-Kohn theorem. Probably you will see many similarities with the video you just watched, but perhaps there is a different emphasis or a different point of view as well. Find out for yourself what are the similarities and differences between these two explanations. Feel free to report your thoughts on the forum (not mandatory, but you need to report yourself before you can see the other forum contributions).
The mathematical proof of this fundamental theorem is surprisingly short and elegant. As an optional item, you can look at the proof here: