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 Th87:
  Y|`R = R /\ [:dom R,Y:]
proof
  set P = R /\ [:dom R,Y:];
  let x,y;
  thus [x,y] in Y|`R implies [x,y] in P
  proof
    assume
A1: [x,y] in Y|`R;
    then
A2: y in Y by Def10;
A3: [x,y] in R by A1,Def10;
    then x in dom R by XTUPLE_0:def 12;
    then [x,y] in [:dom R,Y:] 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 [:dom R,Y:] by XBOOLE_0:def 4; then
A5: y in Y by ZFMISC_1:87;
  [x,y] in R by A4,XBOOLE_0:def 4;
  hence thesis by A5,Def10;
end;
