While the Wikipedia page provide a high-level mathematical explanation of the first Hohenberg-Kohn theorem, it does not emphasize the ultimate bijective nature of the relationship between the exteranl potential and density. In this regard, I found the video explanation from the course to be much more meaningful, especially how it emphasized the one-to-one correspondence between the ground state of a many-electron system and the external potential, which in turn allows us to write any observable ground state property of said system as a unique functional of density.