Anwser First Hohenberg-Kohn theorem

The definition of the first Hohenberg-Kohn theorem that is given on the given Wikipedia page is: “The external potential (and hence the total energy), is a unique functional of the electron density.”
On the wikipedia page they immediately mention the consequence that was mentioned in the video in their own definition. The definition on Wikipedia comes out stronger as a result. In the video it’s mentioned that the density function contains as much information as the wave function, but the statements about the functional are only mentioned at the end of the video.