reserve X,X1,X2,Y,Y1,Y2 for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,g1,g2,h for Function,
  R,S for Relation;
reserve e,u for object,
  A for Subset of X;

theorem
  for f being Function,A being set st f is one-to-one & A c= dom f
  holds f".:(f.:A)=A
proof
  let f be Function,A be set;
  set B = f.:A;
  assume that
A1: f is one-to-one and
A2: A c= dom f;
A3: f".:B c= A
  proof
    let y be object;
    assume y in f".:B;
    then consider x being object such that
    x in dom (f") and
A4: x in B and
A5: y=f".x by Def6;
    ex y2 being object st ( y2 in dom f)&( y2 in A)&( x=f.y2) by A4,Def6;
    hence thesis by A1,A5,Th31;
  end;
  A c= f".:B
  proof
    let x be object;
    assume
A6: x in A;
    set y0=f.x;
A7: f".y0=x by A1,A2,A6,Th33;
    y0 in rng f by A2,A6,Def3;
    then
A8: y0 in dom (f") by A1,Th32;
    y0 in B by A2,A6,Def6;
    hence thesis by A7,A8,Def6;
  end;
  hence thesis by A3;
end;
