Using the cosmological baryonic accretion rate and normal star formation efficiencies, we present a very simple model for star-forming galaxies (SFGs) that accounts for the mass and redshift dependencies of the SFR-Mass and Tully-Fisher relations from z=2 to the present. The time evolution follows from the fact that each modelled galaxy approaches a steady state where the SFR follows the (net) cold gas accretion rate. The key feature of the model is a halo mass floor M_{min}~10^{11} below which accretion is quenched in order to simultaneously account for the observed slopes of the SFR-Mass and Tully-Fischer relations. The same successes cannot be achieved via a star-formation threshold (or delay) nor by varying the SF efficiency or the feedback efficiency. Combined with the mass ceiling for cold accretion due to virial shock heating, the mass floor M_{min} explains galaxy "downsizing", where more massive galaxies formed earlier and over a shorter period of time. It turns out that the model also accounts for the observed galactic baryon and gas fractions as a function of mass and time, and the cosmic SFR density from z~6 to z=0, which are all resulting from the mass floor M_{min}. The model helps to understand that it is the cosmological decline of accretion rate that drives the decrease of cosmic SFR density between z~2 and z=0 and the rise of the cosmic SFR density allows us to put a constraint on our main parameter M_{min}~10^{11} solar masses. Among the physical mechanisms that could be responsible for the mass floor, we view that photo-ionization feedback (from first in-situ hot stars) lowering the cooling efficiency is likely to play a large role. ; Comment: 19pages, 14 figures, accepted to ApJ, updated references