We introduce a family of layer to layer transfer matrices in a three-dimensional (3D) lattice model which can be viewed as partition functions of the q-oscillator valued six-vertex model on an m × n square lattice. By invoking the tetrahedron equation we establish their commutativity and bilinear relations mixing various boundary conditions. At q = 0 and m = n, they ultimately yield a new proof of the steady state formula for the n-species totally asymmetric simple exclusion process obtained recently by the authors, revealing the 3D integrability in the matrix product construction.