Dong, Munczek, and Roberts have shown how the full 3-point vertex that appears in the Schwinger-Dyson equation for the fermion propagator can be expressed in terms of a constrained function W-1 in massless quenched QED. However, this analysis involved two key assumptions: that the fermion anomalous dimension vanishes in the Landau gauge and that the transverse vertex has a simplified dependence on momenta. Here we remove these assumptions and find the general form for a new constrained function U-1 that ensures the multiplicative renormalizability of the fermion propagator nonperturbatively. We then study the restriction imposed on U-1 by recent perturbative calculations of the vertex and compute its leading logarithmic expansion. Since U-1 should reduce to this expansion in the weak coupling regime, this should serve as a guide to its nonperturbative construction. We comment on the perturbative realization of the constraints on U-1.