We investigate analytically a connection between the t-J model and the
strongly correlated Bardeen-Cooper-Schrieffer (BCS) Hamiltonian, with the
effect of strong electron correlations accounted by the Gutzwiller projection.
We show that in the immediate vicinity of half filling the projected 2D BCS
Hamiltonian with strong pairing develops an antiferromagnetically (AF) ordered
ground state. This result explicitly demonstrates that antiferromagnetism in
this model appears as a natural consequence of the strong Coulomb repulsion in
a low doped regime. At moderate doping the ground state of the
Gutzwiller-projected BCS Hamiltonian becomes qualitatively similar to
Anderson's resonating valence bond state which is known to fit nicely the
properties of the t-J model in this regime. These two properties taken together
indicate that the projected BCS Hamiltonian captures the essential low-energy
physics of the t-J model in the whole underdoped region