reserve U for Universe;
reserve x for Element of U;
reserve U1,U2 for Universe;

theorem Th106:
  proj1 Maps U c= [:U,U:] &
  ([:U,U:] \ [: U \ { {} } , { {} } :]) c= proj1 Maps U &
  proj2 Maps U = Funcs U
  proof
    thus proj1 Maps U c= [:U,U:]
    proof
      let o be object;
      assume o in proj1 Maps U;
      then consider o9 be object such that
A1:   [o,o9] in Maps U by XTUPLE_0:def 12;
      consider A,B be Element of U, f be Element of Funcs U such that
A2:   [o,o9] = [[A,B],f] and
      B = {} implies A = {} and
      f is Function of A,B by A1;
      o = [A,B] by A2,XTUPLE_0:1;
      hence o in [:U,U:] by ZFMISC_1:def 2;
    end;
A10: [:U,U:] /\ {[A,B] where A,B is Element of U : (B = {} implies
      A = {})} c= proj1 Maps U
    proof
      let o be object;
      assume
A3:   o in [:U,U:] /\
        {[A,B] where A,B is Element of U : (B = {} implies A = {})};
      then o in [:U,U:] & o in {[A,B] where A,B is Element of U :
        (B = {} implies A = {})} by XBOOLE_0:def 4;
      then consider A,B be object such that
A4:   A in U and
A5:   B in U and
A6:   o = [A,B] by ZFMISC_1:def 2;
      o in {[A,B] where A,B is Element of U : (B = {} implies A = {})}
        by A3,XBOOLE_0:def 4;
      then consider A9,B9 be Element of U such that
A7:   o = [A9,B9] and
A8:   B9 = {} implies A9 = {};
A9:   A = A9 & B = B9 by A6,A7,XTUPLE_0:1;
      reconsider A,B as Element of U by A4,A5;
      set f = the Function of A,B;
      f is Element of Funcs U by ENS_1:1,A8,A9;
      then [[A,B],f] in Maps U by A8,A9;
      hence o in proj1 Maps U by A6,XTUPLE_0:def 12;
    end;
    [:U,U:] /\ ([:U,U:] \ [: U \ { {} } , { {} } :])
      = ([:U,U:] /\ [:U,U:]) \ [: U \ { {} } , { {} } :] by XBOOLE_1:49;
    hence ([:U,U:] \ [: U \ { {} } , { {} } :]) c= proj1 Maps U by A10,Th1;
    proj2 Maps U = Funcs U
    proof
      thus proj2 Maps U c= Funcs U
      proof
        let o be object;
        assume o in proj2 Maps U;
        then consider o9 be object such that
A11:    [o9,o] in Maps U by XTUPLE_0:def 13;
        consider A,B be Element of U, f be Element of Funcs U such that
A12:    [o9,o] = [[A,B],f] and
        B = {} implies A = {} and
        f is Function of A,B by A11;
        o = f by A12,XTUPLE_0:1;
        hence o in Funcs U;
      end;
      let o be object;
      assume
A13:  o in Funcs U;
      then consider A,B be Element of U such that
A14:  B = {} implies A = {} and
A15:  o is Function of A,B by ENS_1:1;
      reconsider f = o as Function of A,B by A15;
      [[A,B],f] in Maps U by A14,A13;
      hence o in proj2 Maps U by XTUPLE_0:def 13;
    end;
    hence thesis;
  end;
