:: ZF_FUND2 semantic presentation
:: deftheorem Def1 defines Section ZF_FUND2:def 1 :
:: deftheorem Def2 defines predicatively_closed ZF_FUND2:def 2 :
theorem Th1: :: ZF_FUND2:1
Lemma4:
for b1 being Function
for b2 being set holds
not ( b2 in Union b1 & ( for b3 being set holds
not ( b3 in dom b1 & b2 in b1 . b3 ) ) )
theorem Th2: :: ZF_FUND2:2
Lemma6:
for b1 being ZF-formula
for b2 being non empty set
for b3 being Function of VAR ,b2
for b4 being Variable holds
( not b4 in variables_in b1 & {(x. 0),(x. 1),(x. 2)} misses Free b1 & b2,b3 |= All (x. 3),(Ex (x. 0),(All (x. 4),(b1 <=> ((x. 4) '=' (x. 0))))) implies ( {(x. 0),(x. 1),(x. 2)} misses Free (b1 / (x. 0),b4) & b2,b3 |= All (x. 3),(Ex (x. 0),(All (x. 4),((b1 / (x. 0),b4) <=> ((x. 4) '=' (x. 0))))) & def_func' b1,b3 = def_func' (b1 / (x. 0),b4),b3 ) )
Lemma7:
for b1, b2 being non empty set
for b3 being Element of b2
for b4 being Element of b1
for b5 being Function of VAR ,b1
for b6 being Function of VAR ,b2
for b7 being Variable holds
( b5 = b6 & b4 = b3 implies b5 / b7,b4 = b6 / b7,b3 )
Lemma8:
for b1 being ZF-formula
for b2 being non empty set
for b3 being Function of VAR ,b2 holds
( b2,b3 |= All (x. 3),(Ex (x. 0),(All (x. 4),(b1 <=> ((x. 4) '=' (x. 0))))) & not x. 4 in Free b1 implies for b4 being Element of b2 holds (def_func' b1,b3) .: b4 = {} )
Lemma9:
for b1 being ZF-formula
for b2, b3 being Variable holds
( not b2 in variables_in b1 & not b3 in variables_in b1 & b2 <> x. 0 & b3 <> x. 0 & b3 <> x. 4 implies ( x. 4 in Free b1 iff x. 0 in Free (Ex (x. 3),(((x. 3) 'in' b2) '&' ((b1 / (x. 0),b3) / (x. 4),(x. 0)))) ) )
theorem Th3: :: ZF_FUND2:3
Lemma11:
for b1 being ZF-formula
for b2 being non empty set
for b3 being Element of b2
for b4 being Function of VAR ,b2
for b5 being Nat holds
( x. b5 in Free b1 implies {[b5,b3]} \/ ((b4 * decode ) | ((code (Free b1)) \ {b5})) = ((b4 / (x. b5),b3) * decode ) | (code (Free b1)) )
theorem Th4: :: ZF_FUND2:4
theorem Th5: :: ZF_FUND2:5
theorem Th6: :: ZF_FUND2:6