The Fermi surface (FS) of Bi2Sr2CaCu2O8+delta (Bi2212) predicted by band theory displays Bi-related pockets around the (pi, 0) point, which have never been observed experimentally. We show that when the effects of hole doping either by substituting Pb for Bi or by adding excess O in Bi2212 are included, the Bi-O bands are lifted above the Fermi energy (E(F)) and the resulting first-principles FS is in remarkable accord with measurements. With decreasing hole doping the Bi-O bands drop below and the system self-dopes below a critical hole concentration. Computations on other Bi- as well as Tl- and Hg-based compounds indicate that lifting of the cation-derived band with hole doping is a general property of the electronic structures of the cuprates.