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 Th32:
  for f being Function,A be set st f is one-to-one holds (.:f).:A c= ("f)"A
proof
  let f be Function,A be set such that
A1: f is one-to-one;
  let y be object;
   reconsider yy=y as set by TARSKI:1;
  assume y in (.:f).:A;
  then consider x being object such that
A2: x in dom(.:f) and
A3: x in A and
A4: y = .:f.x by FUNCT_1:def 6;
  reconsider x as set by TARSKI:1;
A5: x in bool dom f by A2,Def1;
  then
A6: y = f.:x by A4,Def1;
  then
A7: x c= f"yy by A5,FUNCT_1:76;
A8: yy c= rng f by A6,RELAT_1:111;
  then y in bool rng f;
  then
A9: y in dom("f) by Def2;
  f"yy c= x by A1,A6,FUNCT_1:82;
  then f"yy in A by A3,A7,XBOOLE_0:def 10;
  then "f.y in A by A8,Def2;
  hence thesis by A9,FUNCT_1:def 7;
end;
