I consulted the book: Modern Condensed Matter Physics, and on page 386, Theorem 1 states:
“The ground-state of a many-electron system is a functional of the electron density.”
This is consistent with the main idea presented in the video: there is a one-to-one correspondence between the ground-state wavefunction and the electron density.
[1] S. M. Girvin and K. Yang, Modern Condensed Matter Physics. Cambridge: Cambridge University Press, 2019, p. 386. doi: 10.1017/9781316480649.