Article ZF_FUND2, MML version 4.99.1005
:: ZF_FUND2:funcnot 1 => ZF_FUND2:func 1
definition
let a1 be ZF-formula-like FinSequence of NAT;
let a2 be non empty set;
let a3 be Function-like quasi_total Relation of VAR,a2;
func Section(A1,A3) -> Element of bool a2 equals
{b1 where b1 is Element of a2: a2,a3 /(x. 0,b1) |= a1}
if x. 0 in Free a1
otherwise {};
end;
:: ZF_FUND2:def 1
theorem
for b1 being ZF-formula-like FinSequence of NAT
for b2 being non empty set
for b3 being Function-like quasi_total Relation of VAR,b2 holds
(x. 0 in Free b1 implies Section(b1,b3) = {b4 where b4 is Element of b2: b2,b3 /(x. 0,b4) |= b1}) &
(x. 0 in Free b1 or Section(b1,b3) = {});
:: ZF_FUND2:attrnot 1 => ZF_FUND2:attr 1
definition
let a1 be non empty set;
attr a1 is predicatively_closed means
for b1 being ZF-formula-like FinSequence of NAT
for b2 being non empty set
for b3 being Function-like quasi_total Relation of VAR,b2
st b2 in a1
holds Section(b1,b3) in a1;
end;
:: ZF_FUND2:dfs 2
definiens
let a1 be non empty set;
To prove
a1 is predicatively_closed
it is sufficient to prove
thus for b1 being ZF-formula-like FinSequence of NAT
for b2 being non empty set
for b3 being Function-like quasi_total Relation of VAR,b2
st b2 in a1
holds Section(b1,b3) in a1;
:: ZF_FUND2:def 2
theorem
for b1 being non empty set holds
b1 is predicatively_closed
iff
for b2 being ZF-formula-like FinSequence of NAT
for b3 being non empty set
for b4 being Function-like quasi_total Relation of VAR,b3
st b3 in b1
holds Section(b2,b4) in b1;
:: ZF_FUND2:th 1
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Function-like quasi_total Relation of VAR,b1
st b1 is epsilon-transitive
holds Section(All(x. 2,((x. 2) 'in' x. 0) => ((x. 2) 'in' x. 1)),b3 /(x. 1,b2)) = b1 /\ bool b2;
:: ZF_FUND2:th 2
theorem
for b1 being non empty universal set
for b2 being non-empty DOMAIN-yielding T-Sequence of b1
st (for b3, b4 being Ordinal of b1
st b3 in b4
holds b2 . b3 c= b2 . b4) &
(for b3 being Ordinal of b1 holds
b2 . b3 in Union b2 & b2 . b3 is epsilon-transitive) &
Union b2 is predicatively_closed
holds Union b2 |= the_axiom_of_power_sets;
:: ZF_FUND2:th 3
theorem
for b1 being non empty universal set
for b2 being non-empty DOMAIN-yielding T-Sequence of b1
st omega in b1 &
(for b3, b4 being Ordinal of b1
st b3 in b4
holds b2 . b3 c= b2 . b4) &
(for b3 being Ordinal of b1
st b3 <> {} & b3 is being_limit_ordinal
holds b2 . b3 = Union (b2 | b3)) &
(for b3 being Ordinal of b1 holds
b2 . b3 in Union b2 & b2 . b3 is epsilon-transitive) &
Union b2 is predicatively_closed
for b3 being ZF-formula-like FinSequence of NAT
st {x. 0,x. 1,x. 2} misses Free b3
holds Union b2 |= the_axiom_of_substitution_for b3;
:: ZF_FUND2:th 4
theorem
for b1 being ZF-formula-like FinSequence of NAT
for b2 being non empty set
for b3 being Function-like quasi_total Relation of VAR,b2 holds
Section(b1,b3) = {b4 where b4 is Element of b2: {[{},b4]} \/ ((b3 * decode) | ((code Free b1) \ {{}})) in Diagram(b1,b2)};
:: ZF_FUND2:th 5
theorem
for b1 being non empty universal set
for b2 being non empty Element of bool b1
st b2 is closed_wrt_A1-(b1) & b2 is epsilon-transitive
holds b2 is predicatively_closed;
:: ZF_FUND2:th 6
theorem
for b1 being non empty universal set
for b2 being non-empty DOMAIN-yielding T-Sequence of b1
st omega in b1 &
(for b3, b4 being Ordinal of b1
st b3 in b4
holds b2 . b3 c= b2 . b4) &
(for b3 being Ordinal of b1
st b3 <> {} & b3 is being_limit_ordinal
holds b2 . b3 = Union (b2 | b3)) &
(for b3 being Ordinal of b1 holds
b2 . b3 in Union b2 & b2 . b3 is epsilon-transitive) &
Union b2 is closed_wrt_A1-(b1)
holds Union b2 is being_a_model_of_ZF;