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 Th5:
  X <> {} implies for y ex f st dom f = X & rng f = {y}
proof
  assume
A1: X <> {};
  let y;
  deffunc F(object) = y;
  consider f such that
A2: dom f = X and
A3: for x st x in X holds f.x = F(x) from Lambda;
  take f;
  thus dom f = X by A2;
  for y1 be object holds y1 in rng f iff y1 = y
  proof
    let y1 be object;
A4: now
      set x = the Element of X;
      assume
A5:   y1 = y;
      f.x = y by A1,A3;
      hence y1 in rng f by A1,A2,A5,Def3;
    end;
    now
      assume y1 in rng f;
      then ex x being object st x in dom f & y1 = f.x by Def3;
      hence y1 = y by A2,A3;
    end;
    hence thesis by A4;
  end;
  hence thesis by TARSKI:def 1;
end;
