Journal of Formalized Mathematics
Volume 13, 2001
University of Bialystok
Copyright (c) 2001 Association of Mizar Users

### Classes of Independent Partitions

by
Andrzej Trybulec

Received February 14, 2001

MML identifier: PARTIT_2
[ Mizar article, MML identifier index ]

```environ

vocabulary RELAT_1, FUNCT_2, MARGREL1, PARTIT1, EQREL_1, BOOLE, FUNCT_1,
SETFAM_1, CANTOR_1, T_1TOPSP, RELAT_2, ZF_LANG, BVFUNC_1, BVFUNC_2,
FUNCT_4, PARTFUN1;
notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, SETFAM_1, MARGREL1, VALUAT_1,
CANTOR_1, RELAT_1, RELAT_2, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2,
FRAENKEL, FUNCT_4, EQREL_1, PARTIT1, BVFUNC_1, BVFUNC_2;
constructors FUNCT_7, BVFUNC_2, BVFUNC_1, CANTOR_1;
clusters PARTIT1, SUBSET_1, MARGREL1, VALUAT_1, RELSET_1, FUNCT_1, RELAT_1,
EQREL_1, PARTFUN1;
requirements SUBSET, BOOLE;

begin :: Preliminaries

definition let X,Y be set, R,S be Relation of X,Y;
redefine pred R c= S means
:: PARTIT_2:def 1
for x being Element of X, y being Element of Y
holds [x,y] in R implies [x,y] in S;
end;

reserve Y for non empty set,
a for Element of Funcs(Y,BOOLEAN),
G for Subset of PARTITIONS(Y),
P,Q for a_partition of Y;

definition let Y be non empty set, G be non empty Subset of PARTITIONS Y;
redefine mode Element of G -> a_partition of Y;
end;

theorem :: PARTIT_2:1
'/\' {} PARTITIONS Y = %O Y;

theorem :: PARTIT_2:2
for R,S being Equivalence_Relation of Y holds
R \/ S c= R*S;

theorem :: PARTIT_2:3
for R being Relation of Y holds R c= nabla Y;

theorem :: PARTIT_2:4
for R being Equivalence_Relation of Y holds
(nabla Y)*R = nabla Y & R*nabla Y = nabla Y;

theorem :: PARTIT_2:5
for P being a_partition of Y, x,y being Element of Y holds
[x,y] in ERl P iff x in EqClass(y,P);

theorem :: PARTIT_2:6
for P,Q,R being a_partition of Y st ERl(R) = ERl(P)*ERl(Q)
for x,y being Element of Y holds
x in EqClass(y,R) iff
ex z being Element of Y st x in EqClass(z,P) & z in EqClass(y,Q);

theorem :: PARTIT_2:7
for R,S being Relation, Y being set st
R is_reflexive_in Y & S is_reflexive_in Y
holds R*S is_reflexive_in Y;

theorem :: PARTIT_2:8
for R being Relation, Y being set st R is_reflexive_in Y
holds Y c= field R;

theorem :: PARTIT_2:9
for Y being set, R being Relation of Y st R is_reflexive_in Y
holds Y = field R;

theorem :: PARTIT_2:10
for R,S being Equivalence_Relation of Y st R*S = S*R
holds R*S is Equivalence_Relation of Y;

begin :: Boolean-valued Functions

theorem :: PARTIT_2:11
for a,b being Element of Funcs(Y,BOOLEAN) st a '<' b
holds 'not' b '<' 'not' a;

canceled;

theorem :: PARTIT_2:13
for a,b being Element of Funcs(Y,BOOLEAN),
G being Subset of PARTITIONS(Y),
P being a_partition of Y st a '<' b holds
All(a,P,G) '<' All(b,P,G);

canceled;

theorem :: PARTIT_2:15
for a,b being Element of Funcs(Y,BOOLEAN),
G being Subset of PARTITIONS(Y),
P being a_partition of Y st a '<' b holds
Ex(a,P,G) '<' Ex(b,P,G);

begin

theorem :: PARTIT_2:16
G is independent implies
for P,Q being Subset of PARTITIONS Y st P c= G & Q c= G
holds ERl('/\'P)*ERl('/\'Q) = ERl('/\'Q)*ERl('/\'P);

theorem :: PARTIT_2:17
G is independent
implies All(All(a,P,G),Q,G) = All(All(a,Q,G),P,G);

theorem :: PARTIT_2:18
G is independent
implies Ex(Ex(a,P,G),Q,G) = Ex(Ex(a,Q,G),P,G);

theorem :: PARTIT_2:19
for a being Element of Funcs(Y,BOOLEAN),
G being Subset of PARTITIONS(Y),
P,Q being a_partition of Y st G is independent holds
Ex(All(a,P,G),Q,G) '<' All(Ex(a,Q,G),P,G);

```

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