 reserve x,y,z,t for object,X,Y,Z,W for set;
 reserve R,S,T for Relation;

theorem
  R c= id dom R implies R = id dom R
proof
  assume
A1: R c= id dom R;
  let x,y be object;
  thus [x,y] in R implies [x,y] in id dom R by A1;
  assume
A2: [x,y] in id dom R;
  then x in dom R by RELAT_1:def 10; then
A3: ex z being object st [x,z] in R by XTUPLE_0:def 12;
  x = y by A2,RELAT_1:def 10;
  hence thesis by A1,A3,RELAT_1:def 10;
end;
