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 Th29:
  for f being Function holds ("f).:B c= (.:f)"B
proof
  let f be Function;
  let x be object;
   reconsider xx=x as set by TARSKI:1;
  assume x in ("f).:B;
  then consider Y being object such that
A1: Y in dom ("f) and
A2: Y in B and
A3: x = "f.Y by FUNCT_1:def 6;
  reconsider Y as set by TARSKI:1;
A4: "f.Y = f"Y by A1,Th21;
  then
A5: xx c= dom f by A3,RELAT_1:132;
  then x in bool dom f;
  then
A6: x in dom(.:f) by Def1;
  Y in bool rng f by A1,Def2;
  then f.:xx in B by A2,A3,A4,FUNCT_1:77;
  then (.:f).x in B by A5,Def1;
  hence thesis by A6,FUNCT_1:def 7;
end;
