reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;
reserve F,G for Cardinal-Function;
reserve A,B for set;
reserve A,B for Ordinal;
reserve n,k for Nat;

theorem
  for A being non empty countable set
  ex f being Function of omega, A st rng f = A
proof
  let A be non empty countable set;
  consider f being Function such that
A1: dom f = omega and
A2: A c= rng f by Th90;
  consider x being object such that
A3: x in A by XBOOLE_0:def 1;
  set F = f|(f"A) +* (omega \ f"A --> x);
A4: rng F = A & dom F = omega
  proof
A5: f"A c= omega by A1,RELAT_1:132;
A6: dom(f|(f"A)) = omega /\ (f"A) by A1,RELAT_1:61;
    per cases;
    suppose
A7:   omega = f"A;
      then
A8:   omega \ f"A = {} by XBOOLE_1:37;
      then dom(f|(f"A)) /\ dom(omega \ f"A --> x) = {};
      then dom(f|(f"A)) misses dom(omega \ f"A --> x);
      then F = (f|(f"A)) \/ (omega \ f"A --> x) by FUNCT_4:31;
      hence rng F = rng(f|(f"A)) \/ rng(omega \ f"A --> x) by RELAT_1:12
        .= rng(f|(f"A)) \/ {} by A8
        .= f.:(f"A) by RELAT_1:115
        .= A by A2,FUNCT_1:77;
      thus
      dom F = dom(f|(f"A)) \/ dom(omega \ f"A --> x) by FUNCT_4:def 1
        .= dom(f|(f"A)) \/ {} by A8
        .= omega by A6,A7;
    end;
    suppose omega <> f"A; then
A9:  omega \ f"A is non empty by A5,XBOOLE_1:37;
      dom(omega \ f"A --> x) = omega \ f"A;
      then F = (f|(f"A)) \/ (omega \ f"A --> x) by A6,FUNCT_4:31,XBOOLE_1:89;
      hence rng F = rng(f|(f"A)) \/ rng(omega \ f"A --> x) by RELAT_1:12
        .= rng(f|(f"A)) \/ {x} by A9,FUNCOP_1:8
        .= f.:(f"A) \/ {x} by RELAT_1:115
        .= A \/ {x} by A2,FUNCT_1:77
        .= A by A3,ZFMISC_1:40;
      thus
      dom F = dom(f|(f"A)) \/ dom(omega \ f"A --> x) by FUNCT_4:def 1
        .= omega /\ (f"A) \/ (omega \ f"A) by A6
        .= omega by XBOOLE_1:51;
    end;
  end;
  then reconsider F as Function of omega, A by FUNCT_2:def 1,RELSET_1:4;
  take F;
  thus thesis by A4;
end;
