reserve N,M,K for ExtNat;
reserve X for ext-natural-membered set;

theorem Th3:
  for X being set, x being object holds card(X --> x) = card X
proof
  let X be set, x be object;
  deffunc F(object) = [$1, x];
  consider f being Function such that
    A1: dom f = X & for z being object st z in X holds f.z = F(z)
    from FUNCT_1:sch 3;
  now
    let y be object;
    hereby
      assume y in rng f;
      then consider z being object such that
        A2: z in dom f & f.z = y by FUNCT_1:def 3;
      A3: y = [z,x] by A1, A2;
      z in X & x in {x} by A1, A2, TARSKI:def 1;
      then y in [: X, {x} :] by A3, ZFMISC_1:87;
      hence y in X --> x by FUNCOP_1:def 2;
    end;
    assume y in X --> x;
    then consider z,x0 being object such that
      A4: z in X & x0 in {x} & y = [z,x0] by ZFMISC_1:def 2;
    y = [z,x] by A4, TARSKI:def 1
      .= f.z by A1, A4;
    hence y in rng f by A1, A4, FUNCT_1:3;
  end;
  then A5: rng f = X --> x by TARSKI:2;
  now
    let x1,x2 be object;
    assume A6: x1 in dom f & x2 in dom f & f.x1 = f.x2;
    then f.x1 = [x1,x] & f.x2 = [x2,x] by A1;
    hence x1 = x2 by A6, XTUPLE_0:1;
  end;
  hence thesis by A1, A5, FUNCT_1:def 4, CARD_1:70;
end;
