Incomplete beta function and implicit function

Question

Say $\,y\,$ is implicitly given by $$ I_y(\alpha+1,\beta) = \frac{1}{B(\alpha+1,\beta)}\int_0^y t^{\alpha}(1-t)^{\beta-1}dt=k\,. $$ where $\,0<k<1\,$ is a constant. Using Matlab for numerical experiments, I found that $\,I_y(\alpha,\beta)\,$ is decreasing in $\alpha$, can we prove it analytically?