Article ZFMODEL2, MML version 4.99.1005

:: ZFMODEL2:th 1
theorem
for b1, b2 being Element of VAR
for b3 being ZF-formula-like FinSequence of NAT holds
   Free (b3 /(b1,b2)) c= ((Free b3) \ {b1}) \/ {b2};

:: ZFMODEL2:th 2
theorem
for b1, b2 being Element of VAR
for b3 being ZF-formula-like FinSequence of NAT
      st not b1 in variables_in b3
   holds (b2 in Free b3 implies Free (b3 /(b2,b1)) = ((Free b3) \ {b2}) \/ {b1}) &
    (b2 in Free b3 or Free (b3 /(b2,b1)) = Free b3);

:: ZFMODEL2:th 3
theorem
for b1 being ZF-formula-like FinSequence of NAT holds
   variables_in b1 is finite;

:: ZFMODEL2:th 4
theorem
for b1 being ZF-formula-like FinSequence of NAT holds
   (ex b2 being Element of NAT st
       for b3 being Element of NAT
             st x. b3 in variables_in b1
          holds b3 < b2) &
    (ex b2 being Element of VAR st
       not b2 in variables_in b1);

:: ZFMODEL2:th 5
theorem
for b1 being Element of VAR
for b2 being non empty set
for b3 being ZF-formula-like FinSequence of NAT
for b4 being Function-like quasi_total Relation of VAR,b2
      st not b1 in variables_in b3
   holds    b2,b4 |= b3
   iff
      b2,b4 |= All(b1,b3);

:: ZFMODEL2:th 6
theorem
for b1 being Element of VAR
for b2 being non empty set
for b3 being Element of b2
for b4 being ZF-formula-like FinSequence of NAT
for b5 being Function-like quasi_total Relation of VAR,b2
      st not b1 in variables_in b4
   holds    b2,b5 |= b4
   iff
      b2,b5 /(b1,b3) |= b4;

:: ZFMODEL2:th 7
theorem
for b1, b2, b3 being Element of VAR
for b4 being non empty set
for b5, b6, b7 being Element of b4
for b8 being Function-like quasi_total Relation of VAR,b4
      st b1 <> b2 & b2 <> b3 & b3 <> b1
   holds ((b8 /(b1,b5)) /(b2,b6)) /(b3,b7) = ((b8 /(b3,b7)) /(b2,b6)) /(b1,b5) &
    ((b8 /(b1,b5)) /(b2,b6)) /(b3,b7) = ((b8 /(b2,b6)) /(b3,b7)) /(b1,b5);

:: ZFMODEL2:th 8
theorem
for b1, b2, b3, b4 being Element of VAR
for b5 being non empty set
for b6, b7, b8, b9 being Element of b5
for b10 being Function-like quasi_total Relation of VAR,b5
      st b1 <> b2 & b1 <> b3 & b1 <> b4 & b2 <> b3 & b2 <> b4 & b3 <> b4
   holds (((b10 /(b1,b6)) /(b2,b7)) /(b3,b8)) /(b4,b9) = (((b10 /(b2,b7)) /(b3,b8)) /(b4,b9)) /(b1,b6) &
    (((b10 /(b1,b6)) /(b2,b7)) /(b3,b8)) /(b4,b9) = (((b10 /(b3,b8)) /(b4,b9)) /(b1,b6)) /(b2,b7) &
    (((b10 /(b1,b6)) /(b2,b7)) /(b3,b8)) /(b4,b9) = (((b10 /(b4,b9)) /(b2,b7)) /(b3,b8)) /(b1,b6);

:: ZFMODEL2:th 9
theorem
for b1, b2, b3, b4 being Element of VAR
for b5 being non empty set
for b6, b7, b8, b9, b10 being Element of b5
for b11 being Function-like quasi_total Relation of VAR,b5 holds
   ((b11 /(b1,b6)) /(b2,b7)) /(b1,b8) = (b11 /(b2,b7)) /(b1,b8) &
    (((b11 /(b1,b6)) /(b2,b7)) /(b3,b9)) /(b1,b8) = ((b11 /(b2,b7)) /(b3,b9)) /(b1,b8) &
    ((((b11 /(b1,b6)) /(b2,b7)) /(b3,b9)) /(b4,b10)) /(b1,b8) = (((b11 /(b2,b7)) /(b3,b9)) /(b4,b10)) /(b1,b8);

:: ZFMODEL2:th 10
theorem
for b1 being Element of VAR
for b2 being non empty set
for b3 being Element of b2
for b4 being ZF-formula-like FinSequence of NAT
for b5 being Function-like quasi_total Relation of VAR,b2
      st not b1 in Free b4
   holds    b2,b5 |= b4
   iff
      b2,b5 /(b1,b3) |= b4;

:: ZFMODEL2:th 11
theorem
for b1 being non empty set
for b2 being ZF-formula-like FinSequence of NAT
for b3 being Function-like quasi_total Relation of VAR,b1
   st not x. 0 in Free b2 &
      b1,b3 |= All(x. 3,Ex(x. 0,All(x. 4,b2 <=> ((x. 4) '=' x. 0))))
for b4, b5 being Element of b1 holds
   (def_func'(b2,b3)) . b4 = b5
iff
   b1,(b3 /(x. 3,b4)) /(x. 4,b5) |= b2;

:: ZFMODEL2:th 12
theorem
for b1 being non empty set
for b2 being ZF-formula-like FinSequence of NAT
for b3 being Function-like quasi_total Relation of VAR,b1
      st Free b2 c= {x. 3,x. 4} &
         b1 |= All(x. 3,Ex(x. 0,All(x. 4,b2 <=> ((x. 4) '=' x. 0))))
   holds def_func'(b2,b3) = def_func(b2,b1);

:: ZFMODEL2:th 13
theorem
for b1, b2 being Element of VAR
for b3 being non empty set
for b4 being ZF-formula-like FinSequence of NAT
for b5 being Function-like quasi_total Relation of VAR,b3
      st not b1 in variables_in b4
   holds    b3,b5 |= b4 /(b2,b1)
   iff
      b3,b5 /(b2,b5 . b1) |= b4;

:: ZFMODEL2:th 14
theorem
for b1, b2 being Element of VAR
for b3 being non empty set
for b4 being ZF-formula-like FinSequence of NAT
for b5 being Function-like quasi_total Relation of VAR,b3
      st not b1 in variables_in b4 & b3,b5 |= b4
   holds b3,b5 /(b1,b5 . b2) |= b4 /(b2,b1);

:: ZFMODEL2:th 15
theorem
for b1, b2 being Element of VAR
for b3 being non empty set
for b4 being ZF-formula-like FinSequence of NAT
for b5 being Function-like quasi_total Relation of VAR,b3
      st not x. 0 in Free b4 &
         b3,b5 |= All(x. 3,Ex(x. 0,All(x. 4,b4 <=> ((x. 4) '=' x. 0)))) &
         not b1 in variables_in b4 &
         b2 <> x. 3 &
         b2 <> x. 4 &
         not b2 in Free b4 &
         b1 <> x. 0 &
         b1 <> x. 3 &
         b1 <> x. 4
   holds not x. 0 in Free (b4 /(b2,b1)) &
    b3,b5 /(b1,b5 . b2) |= All(x. 3,Ex(x. 0,All(x. 4,(b4 /(b2,b1)) <=> ((x. 4) '=' x. 0)))) &
    def_func'(b4,b5) = def_func'(b4 /(b2,b1),b5 /(b1,b5 . b2));

:: ZFMODEL2:th 16
theorem
for b1, b2 being Element of VAR
for b3 being non empty set
for b4 being ZF-formula-like FinSequence of NAT
      st not b1 in variables_in b4
   holds    b3 |= b4 /(b2,b1)
   iff
      b3 |= b4;

:: ZFMODEL2:th 17
theorem
for b1 being non empty set
for b2 being Element of NAT
for b3 being ZF-formula-like FinSequence of NAT
for b4 being Function-like quasi_total Relation of VAR,b1
      st not x. 0 in Free b3 &
         b1,b4 |= All(x. 3,Ex(x. 0,All(x. 4,b3 <=> ((x. 4) '=' x. 0))))
   holds ex b5 being ZF-formula-like FinSequence of NAT st
      ex b6 being Function-like quasi_total Relation of VAR,b1 st
         (for b7 being Element of NAT
                st b7 < b2 & x. b7 in variables_in b5 & b7 <> 3
             holds b7 = 4) &
          not x. 0 in Free b5 &
          b1,b6 |= All(x. 3,Ex(x. 0,All(x. 4,b5 <=> ((x. 4) '=' x. 0)))) &
          def_func'(b3,b4) = def_func'(b5,b6);

:: ZFMODEL2:th 18
theorem
for b1 being non empty set
for b2 being ZF-formula-like FinSequence of NAT
for b3 being Function-like quasi_total Relation of VAR,b1
      st not x. 0 in Free b2 &
         b1,b3 |= All(x. 3,Ex(x. 0,All(x. 4,b2 <=> ((x. 4) '=' x. 0))))
   holds ex b4 being ZF-formula-like FinSequence of NAT st
      ex b5 being Function-like quasi_total Relation of VAR,b1 st
         (Free b2) /\ Free b4 c= {x. 3,x. 4} &
          not x. 0 in Free b4 &
          b1,b5 |= All(x. 3,Ex(x. 0,All(x. 4,b4 <=> ((x. 4) '=' x. 0)))) &
          def_func'(b2,b3) = def_func'(b4,b5);

:: ZFMODEL2:th 19
theorem
for b1 being non empty set
for b2, b3 being Relation-like Function-like set
      st b2 is_definable_in b1 & b3 is_definable_in b1
   holds b3 * b2 is_definable_in b1;

:: ZFMODEL2:th 20
theorem
for b1 being non empty set
for b2 being ZF-formula-like FinSequence of NAT
for b3 being Function-like quasi_total Relation of VAR,b1
      st not x. 0 in Free b2
   holds    b1,b3 |= All(x. 3,Ex(x. 0,All(x. 4,b2 <=> ((x. 4) '=' x. 0))))
   iff
      for b4 being Element of b1 holds
         ex b5 being Element of b1 st
            for b6 being Element of b1 holds
                  b1,(b3 /(x. 3,b4)) /(x. 4,b6) |= b2
               iff
                  b6 = b5;

:: ZFMODEL2:th 21
theorem
for b1 being non empty set
for b2 being ZF-formula-like FinSequence of NAT
for b3, b4 being Relation-like Function-like set
   st b3 is_definable_in b1 & b4 is_definable_in b1 & Free b2 c= {x. 3}
for b5 being Relation-like Function-like set
      st proj1 b5 = b1 &
         (for b6 being Function-like quasi_total Relation of VAR,b1 holds
            (b1,b6 |= b2 implies b5 . (b6 . x. 3) = b3 . (b6 . x. 3)) &
             (b1,b6 |= 'not' b2 implies b5 . (b6 . x. 3) = b4 . (b6 . x. 3)))
   holds b5 is_definable_in b1;

:: ZFMODEL2:th 22
theorem
for b1 being non empty set
for b2, b3 being Relation-like Function-like set
      st b2 is_parametrically_definable_in b1 & b3 is_parametrically_definable_in b1
   holds b2 * b3 is_parametrically_definable_in b1;

:: ZFMODEL2:th 23
theorem
for b1 being non empty set
for b2, b3, b4 being ZF-formula-like FinSequence of NAT
for b5 being Function-like quasi_total Relation of VAR,b1
   st {x. 0,x. 1,x. 2} misses Free b2 &
      b1,b5 |= All(x. 3,Ex(x. 0,All(x. 4,b2 <=> ((x. 4) '=' x. 0)))) &
      {x. 0,x. 1,x. 2} misses Free b3 &
      b1,b5 |= All(x. 3,Ex(x. 0,All(x. 4,b3 <=> ((x. 4) '=' x. 0)))) &
      {x. 0,x. 1,x. 2} misses Free b4 &
      not x. 4 in Free b4
for b6 being Relation-like Function-like set
      st proj1 b6 = b1 &
         (for b7 being Element of b1 holds
            (b1,b5 /(x. 3,b7) |= b4 implies b6 . b7 = (def_func'(b2,b5)) . b7) &
             (b1,b5 /(x. 3,b7) |= 'not' b4 implies b6 . b7 = (def_func'(b3,b5)) . b7))
   holds b6 is_parametrically_definable_in b1;

:: ZFMODEL2:th 24
theorem
for b1 being non empty set holds
   id b1 is_definable_in b1;

:: ZFMODEL2:th 25
theorem
for b1 being non empty set holds
   id b1 is_parametrically_definable_in b1;