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
  for f being Function st f is one-to-one holds ("f).:B = (.:f)"B
proof
  let f be Function such that
A1: f is one-to-one;
A2: (.:f)"B c= ("f).:B
  proof
    let x be object;
   reconsider xx=x as set by TARSKI:1;
A3: f.:xx c= rng f by RELAT_1:111;
    then f.:xx in bool rng f;
    then
A4: f.:xx in dom("f) by Def2;
    assume
A5: x in (.:f)"B;
    then
A6: x in dom(.:f) by FUNCT_1:def 7;
    then x in bool dom f by Def1;
    then
A7: xx c= f"(f.:xx) by FUNCT_1:76;
    f"(f.:xx) c= xx by A1,FUNCT_1:82;
    then x = f"(f.:xx) by A7,XBOOLE_0:def 10;
    then
A8: x =("f).(f.:xx) by A3,Def2;
    (.:f).x in B by A5,FUNCT_1:def 7;
    then f.:xx in B by A6,Th7;
    hence thesis by A8,A4,FUNCT_1:def 6;
  end;
  ("f).:B c= (.:f)"B by Th29;
  hence thesis by A2,XBOOLE_0:def 10;
end;
