:: Mostowski's Fundamental Operations - Part II
:: by Grzegorz Bancerek and Andrzej Kondracki
::
:: Received February 15, 1991
:: Copyright (c) 1991-2021 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, ZF_LANG, XBOOLE_0, SUBSET_1, FUNCT_1, ARYTM_3, CARD_1,
ZF_MODEL, TARSKI, ORDINAL1, BVFUNC_2, XBOOLEAN, ZFMISC_1, CLASSES2,
ZF_REFLE, CARD_3, RELAT_1, ORDINAL2, ZF_LANG1, ZFMODEL1, XXREAL_0,
REALSET1, ZF_FUND1, FUNCT_2, ORDINAL4, NAT_1, ZF_FUND2;
notations TARSKI, XBOOLE_0, ENUMSET1, ZFMISC_1, SUBSET_1, CARD_1, ORDINAL1,
XCMPLX_0, NAT_1, RELAT_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, ZF_LANG,
ZF_MODEL, ZFMODEL1, ORDINAL2, NUMBERS, CARD_3, ZF_LANG1, CLASSES2,
ORDINAL4, ZF_REFLE, ZF_FUND1, XXREAL_0;
constructors PARTFUN1, WELLORD2, DOMAIN_1, XXREAL_0, XREAL_0, NAT_1, CLASSES1,
ZFMODEL1, ZF_LANG1, ZF_REFLE, ZF_FUND1, ORDINAL4, RELSET_1, ZF_MODEL,
NUMBERS;
registrations ORDINAL1, RELSET_1, FINSET_1, XREAL_0, CARD_1, CLASSES2,
ZF_LANG, ORDINAL4, FUNCT_1, ZF_FUND1, ZF_LANG1;
requirements NUMERALS, REAL, SUBSET, BOOLE, ARITHM;
begin
reserve H for ZF-formula,
M,E for non empty set,
e for Element of E,
m,m0,m3, m4 for Element of M,
v,v1,v2 for Function of VAR,M,
f,f1 for Function of VAR,E,
g for Function,
u,u1,u2 for set,
x,y for Variable,
i,n for Element of NAT,
X for set;
definition
let H,M,v;
func Section(H,v) -> Subset of M equals
:: ZF_FUND2:def 1
{ m : M,v/(x.0,m) |= H } if
x.0 in Free H otherwise {};
end;
definition
let M;
attr M is predicatively_closed means
:: ZF_FUND2:def 2
for H, E, f st E in M holds Section(H,f) in M;
end;
theorem :: ZF_FUND2:1
E is epsilon-transitive implies Section(All(x.2,x.2 'in' x.0 =>
x.2 'in' x.1),f/(x.1,e)) = E /\ bool e;
reserve W for Universe,
w for Element of W,
Y for Subclass of W,
a,a1,b,c for Ordinal of W,
L for DOMAIN-Sequence of W;
theorem :: ZF_FUND2:2
(for a,b st a in b holds L.a c= L.b) & (for a holds L.a in Union
L & L.a is epsilon-transitive) & Union L is predicatively_closed implies Union
L |= the_axiom_of_power_sets;
theorem :: ZF_FUND2:3
omega in W & (for a,b st a in b holds L.a c= L.b) & (for a st a
<> {} & a is limit_ordinal holds L.a = Union (L|a)) & (for a holds L.a in Union
L & L.a is epsilon-transitive) & Union L is predicatively_closed implies for H
st {x.0,x.1,x.2} misses Free H holds Union L |= the_axiom_of_substitution_for H
;
theorem :: ZF_FUND2:4
Section(H,v)= {m : {[{},m]} \/ (v*decode)|((code Free H)\{{}}) in
Diagram(H,M)};
theorem :: ZF_FUND2:5
Y is closed_wrt_A1-A7 & Y is epsilon-transitive implies Y is
predicatively_closed;
theorem :: ZF_FUND2:6
omega in W & (for a,b st a in b holds L.a c= L.b) & (for a st a <> {}
& a is limit_ordinal holds L.a = Union (L|a)) & (for a holds L.a in Union L & L
.a is epsilon-transitive) & Union L is closed_wrt_A1-A7 implies Union L is
being_a_model_of_ZF;