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;

theorem
  f is one-to-one implies f.:X = (f")"X
proof
  assume
A1: f is one-to-one;
  for y being object holds y in f.:X iff y in (f")"X
  proof let y be object;
    thus y in f.:X implies y in (f")"X
    proof
      assume y in f.:X;
      then consider x being object such that
A2:   x in dom f and
A3:   x in X and
A4:   y = f.x by Def6;
      y in rng f by A2,A4,Def3;
      then
A5:   y in dom(f") by A1,Th31;
      (f").(f.x) = x by A1,A2,Th31;
      hence thesis by A3,A4,A5,Def7;
    end;
    assume
A6: y in (f")"X;
    then
A7: (f").y in X by Def7;
    y in dom(f") by A6,Def7;
    then y in rng(f) by A1,Th31;
    then consider x being object such that
A8: x in dom(f) & y = f.x by Def3;
    (f").y = x by A1,A8,Th33;
    hence thesis by A7,A8,Def6;
  end;
  hence thesis by TARSKI:2;
end;
