reserve p,q,x,x1,x2,y,y1,y2,z,z1,z2 for set;
reserve A,B,V,X,X1,X2,Y,Y1,Y2,Z for set;
reserve C,C1,C2,D,D1,D2 for non empty set;

theorem Th20:
  for f being Function of X,Y st Y = {} implies X = {} holds .:f
  is Function of bool X, bool Y
proof
  let f be Function of X,Y;
  assume Y = {} implies X = {};
  then
A1: dom f = X by FUNCT_2:def 1;
  rng f c= Y by RELAT_1:def 19;
  then
A2: bool rng f c= bool Y by ZFMISC_1:67;
A3: .:f is Function of bool dom f, bool rng f by Th19;
  then rng.:f c= bool rng f by RELAT_1:def 19;
  then
A4: rng.:f c= bool Y by A2;
  dom.:f = bool dom f by A3,FUNCT_2:def 1;
  hence thesis by A1,A4,FUNCT_2:def 1,RELSET_1:4;
end;
