This simple code is working fine:

x =.; y =.; z =.;

syst = {Abs[x - y] <= 1, Abs[y - z] <= 1, Abs[z - x] <= 1};

RegionPlot3D[Apply[And, syst], {x, -2, 2}, {y, -2, 2}, {z, -2, 2}]

Result is like this:

I want to draw a crosssection of this volume with the plane z==0.

In my opinion, the RegionFunction is a good way, but

RegionPlot3D[Apply[And,syst],{x,-2,2},{y,-2,2},{z,-2,2},RegionFunction->Function[{x,y,z},z==0]]

returns

RegionPlot3D::optx: "Unknown option RegionFunction->Function[{x,y,z},z==0] in RegionPlot3D[And@@syst,{x,-2,2},{y,-2,2},{z,-2,2},RegionFunction->Function[{x,y,z},z==0]]."

It seems, that RegionFunction is not allowed in RegionPlot3D. But in the help for RegionPlot3D is written

*The arguments supplied to functions in MeshFunctions and RegionFunction are x, y, and z. *Please, where is the problem?

Thanks!