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

theorem Th1:
  dom (R /\ [:X,Y:]) c= X & rng (R /\ [:X,Y:]) c= Y
proof
    per cases;
    suppose
      X = {} or Y = {};
      then R /\ [:X,Y:] = R /\ {} by ZFMISC_1:90
        .= {};
      hence thesis;
    end;
    suppose
A1:   X <> {} & Y <> {};
      rng (R /\ [:X,Y:]) c= rng R /\ rng [:X,Y:] by RELAT_1:13; then
A2:   rng (R /\ [:X,Y:]) c= rng R /\ Y by A1,Lm1;
      dom (R /\ [:X,Y:]) c= dom R /\ dom [:X,Y:] by XTUPLE_0:24; then
A3:   dom (R /\ [:X,Y:]) c= dom R /\ X by A1,Lm1;
      dom R /\ X c= X by XBOOLE_1:17;
      hence dom (R /\ [:X,Y:]) c= X by A3;
      rng R /\ Y c= Y by XBOOLE_1:17;
      hence rng (R /\ [:X,Y:]) c= Y by A2;
    end;
end;
