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 Th61:
  R|X = R /\ [:X,rng R:]
proof
  set P = R /\ [:X,rng R:];
  let x,y;
  thus [x,y] in R|X implies [x,y] in P
  proof
    assume
A1: [x,y] in R|X;
    then
A2: x in X by Def9;
A3: [x,y] in R by A1,Def9;
    then y in rng R by XTUPLE_0:def 13;
    then [x,y] in [:X,rng R:] by A2,ZFMISC_1:def 2;
    hence thesis by A3,XBOOLE_0:def 4;
  end;
  assume
A4: [x,y] in P;
  then [x,y] in [:X,rng R:] by XBOOLE_0:def 4;
  then
A5: x in X by ZFMISC_1:87;
  [x,y] in R by A4,XBOOLE_0:def 4;
  hence thesis by A5,Def9;
end;
