:: Families of Subsets
:: by Andrzej Trybulec
::
:: Received December 13, 2012
:: Copyright (c) 2012-2021 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 MCART_1, RECDEF_2;
notations TARSKI, XTUPLE_0;
constructors TARSKI, XTUPLE_0;
begin
scheme :: XFAMILY:sch 1
Separation { A()-> set, P[object] } :
ex X being set st for x being set holds x in X iff x in A() & P[x];
scheme :: XFAMILY:sch 2
Extensionality { X,Y() -> set, P[set] } : X() = Y()
provided
for x being set holds x in X() iff P[x] and
for x being set holds x in Y() iff P[x];
scheme :: XFAMILY:sch 3
SetEq { P[set] } :
for X1,X2 being set st
(for x being set holds x in X1 iff P[x]) &
(for x being set holds x in X2 iff P[x]) holds X1 = X2;
definition let x be object;
redefine func x`1 -> set;
redefine func x`2 -> set;
end;
definition
let x be object;
redefine func x`1_3 -> set;
redefine func x`2_3 -> set;
end;
definition
let x be object;
redefine func x`1_4 -> set;
redefine func x`2_4 -> set;
end;
:: definition
:: let x1,x2 be element;
:: redefine func [x1,x2] -> set;
:: coherence by TARSKI:1;
:: end;
:: definition
:: let x1,x2,x3 be element;
:: redefine func [x1,x2,x3] -> set;
:: coherence by TARSKI:1;
:: end;