  We study magnetohydrodynamic (MHD) effects arising in the collapse of
magnetized, rotating, massive stellar cores to proto-neutron stars (PNSs). We
perform axisymmetric numerical simulations in full general relativity with a
hybrid equation of state. The formation and early evolution of a PNS are
followed with a grid of 2500 x 2500 zones, which provides better resolution
than in previous (Newtonian) studies. We confirm that significant differential
rotation results even when the rotation of the progenitor is initially uniform.
Consequently, the magnetic field is amplified both by magnetic winding and the
magnetorotational instability (MRI). Even if the magnetic energy E_EM is much
smaller than the rotational kinetic energy T_rot at the time of PNS formation,
the ratio E_EM/T_rot increases to 0.1-0.2 by the magnetic winding. Following
PNS formation, MHD outflows lead to losses of rest mass, energy, and angular
momentum from the system. The earliest outflow is produced primarily by the
increasing magnetic stress caused by magnetic winding. The MRI amplifies the
poloidal field and increases the magnetic stress, causing further angular
momentum transport and helping to drive the outflow. After the magnetic field
saturates, a nearly stationary, collimated magnetic field forms near the
rotation axis and a Blandford-Payne type outflow develops along the field
lines. These outflows remove angular momentum from the PNS at a rate given by
\dot{J} \sim \eta E_EM C_B, where \eta is a constant of order 0.1 and C_B is a
typical ratio of poloidal to toroidal field strength. As a result, the rotation
period quickly increases for a strongly magnetized PNS until the degree of
differential rotation decreases. Our simulations suggest that rapidly rotating,
magnetized PNSs may not give rise to rapidly rotating neutron stars.
