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We analyse measurements of the evolving stellar mass (M0) at which the bending of the star-forming main sequence (MS) occurs over 0<z<4. We find M0~10^10Msun over 0<z<1, then M0 rises up to ~10^11Msun at z=2, and then stays flat or slowly increases towards higher redshifts. When converting M0 values into hosting dark matter halo masses, we show that this behaviour is remarkably consistent with the evolving cold- to hot-accretion transition mass, as predicted by theory and defined by the redshift-independent Mshock at z<1.4 and by the rising Mstream at z>1.4 (for which we propose a revision in agreement with latest simulations). We hence argue that the MS bending is primarily due to the lessening of cold-accretion causing a reduction in available cold gas in galaxies and supports predictions of gas feeding theory. In particular, the rapidly rising M0 with redshift at z>1 is confirming evidence for the cold-streams scenario. In this picture, a progressive fueling reduction rather than its sudden suppression in halos more massive than Mshock/Mstream produces a nearly constant star-formation rate in galaxies with stellar masses larger than M0, and not their quenching, for which other physical processes are thus required. Compared to the knee M* in the stellar mass function of galaxies, M0 is significantly lower at z<1.5, and higher at z>2, suggesting that the imprint of gas deprivation on the distribution of galaxy masses happened at early times (z>1.5-2). The typical mass at which galaxies inside the MS become bulge-dominated evolves differently from M0, consistent with the idea that bulge-formation is a distinct process from the phasing-out of cold-accretion.