reserve A, B, X, Y for set;

theorem Th4:
  dom <:pr2(X,Y),pr1(X,Y):> = [:X,Y:] & rng <:pr2(X,Y),pr1(X,Y):> = [:Y,X:]
proof
  set f = <:pr2(X,Y),pr1(X,Y):>;
  thus
A1: dom f = dom pr2(X,Y) /\ dom pr1(X,Y) by FUNCT_3:def 7
    .= dom pr2(X,Y) /\ [:X,Y:] by FUNCT_3:def 4
    .= [:X,Y:] /\ [:X,Y:] by FUNCT_3:def 5
    .= [:X,Y:];
  rng f c= [:rng pr2(X,Y),rng pr1(X,Y):] by FUNCT_3:51;
  hence rng f c= [:Y,X:] by XBOOLE_1:1;
  let y be object;
  assume y in [:Y,X:];
  then consider y1, y2 being object such that
A2: y1 in Y & y2 in X and
A3: y = [y1,y2] by ZFMISC_1:def 2;
A4: [y2,y1] in dom f by A1,A2,ZFMISC_1:87;
  f.(y2,y1) = y by A2,A3,Lm1;
  hence thesis by A4,FUNCT_1:def 3;
end;
