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 Th31:
  for f being Function,A be set st A c= bool dom f holds ("f)"A c= (.:f).:A
proof
  let f be Function,A be set such that
A1: A c= bool dom f;
  let y be object;
   reconsider yy=y as set by TARSKI:1;
  assume
A2: y in ("f)"A;
  then
A3: "f.y in A by FUNCT_1:def 7;
  y in dom("f) by A2,FUNCT_1:def 7;
  then
A4: y in bool rng f by Def2;
  then
A5: f"yy in A by A3,Def2;
  then f"yy in bool dom f by A1;
  then
A6: f"yy in dom .:f by Def1;
  f.:(f"yy) = y by A4,FUNCT_1:77;
  then
A7: .:f.(f"yy) = y by A1,A5,Def1;
  f"yy in A by A3,A4,Def2;
  hence thesis by A7,A6,FUNCT_1:def 6;
end;
