reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

theorem
  (dom R) /\ X c= (R~).:(R.:X)
proof
  let x be object;
  assume
A1: x in (dom R) /\ X;
  then x in dom R by XBOOLE_0:def 4;
  then consider y being object such that
A2: [x,y] in R by XTUPLE_0:def 12;
A3: [y,x] in R~ by A2,Def5;
  x in X by A1,XBOOLE_0:def 4;
  then y in R.:X by A2,Def11;
  hence thesis by A3,Def11;
end;
