We present the Galaxy Stellar Mass Function (MF) up to z~1 from the zCOSMOS-bright 10k spectroscopic sample. We investigate the total MF and the contribution of ETGs and LTGs, defined by different criteria (SED, morphology or star formation). We unveil a galaxy bimodality in the global MF, better represented by 2 Schechter functions dominated by ETGs and LTGs, respectively. For the global population we confirm that low-mass galaxies number density increases later and faster than for massive galaxies. We find that the MF evolution at intermediate-low values of Mstar (logM<10.6) is mostly explained by the growth in stellar mass driven by smoothly decreasing star formation activities. The low residual evolution is consistent with ~0.16 merger per galaxy per Gyr (of which fewer than 0.1 are major). We find that ETGs increase in number density with cosmic time faster for decreasing Mstar, with a median building redshift increasing with mass, in contrast with hierarchical models. For LTGs we find that the number density of blue or spiral galaxies remains almost constant from z~1. Instead, the most extreme population of active star forming galaxies is rapidly decreasing in number density. We suggest a transformation from blue active spirals of intermediate mass into blue quiescent and successively (1-2 Gyr after) into red passive types. The complete morphological transformation into red spheroidals, required longer time-scales or follows after 1-2 Gyr. A continuous replacement of blue galaxies is expected by low-mass active spirals growing in stellar mass. We estimate that on average ~25% of blue galaxies is transforming into red per Gyr for logM<11. We conclude that the build-up of galaxies and ETGs follows the same downsizing trend with mass as the formation of their stars, converse to the trend predicted by current SAMs. We expect a negligible evolution of the global Galaxy Baryonic MF.
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