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