In Density Functional Theory, the 2nd theorem of Hohenberg and Kohn states:
The electron density of the system that minimises the ground-state energy is the exact electron density of the system.
Similarly, there is also the widely-known Variational Principle in quantum chemistry which states:
The wavefunction that minimises the ground-state energy of the system is the exact ground-state wavefunction.
They seem very similar to me and I would just like to ask if there are any similarities between the two, perhaps in terms of the proofs of these two fundamental theorems.