reserve V, C for set;
reserve A, B, D for Element of Fin PFuncs (V, C);
reserve s for Element of PFuncs (V,C);

theorem Th10:
  for b be finite set holds b in B implies ex c be set st c c= b & c in mi B
proof
  defpred X[set,set] means $1 c= $2;
  let b be finite set;
  assume
A1: b in B;
A2: B c= PFuncs (V,C) by FINSUB_1:def 5;
  then reconsider b9 = b as Element of PFuncs (V, C) by A1;
A3: for a,b,c be Element of PFuncs (V, C) st X[a,b] & X[b,c] holds X[a,c] by
XBOOLE_1:1;
A4: for a be Element of PFuncs (V, C) holds X[a,a];
  for a be Element of PFuncs (V, C) st a in B ex b be Element of PFuncs (V
, C) st b in B & X[b,a] & for c be Element of PFuncs (V, C) st c in B & X[c,b]
  holds X[b,c] from FRAENKEL:sch 23(A4,A3);
  then consider c be Element of PFuncs (V, C) such that
A5: c in B and
A6: c c= b9 and
A7: for a be Element of PFuncs (V, C) st a in B & a c= c holds c c= a by A1;
  take c;
  thus c c= b by A6;
  for b be finite set st b in B & b c= c holds b = c by A2,A7;
  hence thesis by A5,A6,Th7;
end;
