CLASSES1: Tarski's Classes and Ranks

:: Tarski's Classes and Ranks
:: by Grzegorz Bancerek
::
:: Received March 23, 1990
:: Copyright (c) 1990-2021 Association of Mizar Users

definition

let B be set ;

attr B is subset-closed means :Def1: :: CLASSES1:def 1
for X, Y being set st X in B & Y c= X holds
Y in B;

end;

:: deftheorem Def1 defines subset-closed CLASSES1:def 1 :

definition

let B be set ;

attr B is Tarski means :: CLASSES1:def 2
( B is subset-closed & ( for X being set st X in B holds
bool X in B ) & ( for X being set holds
( not X c= B or X,B are_equipotent or X in B ) ) );

end;

:: deftheorem defines Tarski CLASSES1:def 2 :

definition

let A, B be set ;

pred B is_Tarski-Class_of A means :: CLASSES1:def 3
( A in B & B is Tarski );

end;

:: deftheorem defines is_Tarski-Class_of CLASSES1:def 3 :

definition

let A be set ;

func Tarski-Class A -> set means :Def4: :: CLASSES1:def 4
( it is_Tarski-Class_of A & ( for D being set st D is_Tarski-Class_of A holds
it c= D ) );

existence

proof end;

uniqueness ;

end;

:: deftheorem Def4 defines Tarski-Class CLASSES1:def 4 :

registration

let A be set ;

cluster Tarski-Class A -> non empty ;

coherence

proof end;

end;

theorem :: CLASSES1:1

for W being set holds
( W is Tarski iff ( W is subset-closed & ( for X being set st X in W holds
bool X in W ) & ( for X being set st X c= W & card X in card W holds
X in W ) ) )

proof end;

theorem Th2: :: CLASSES1:2

for X being set holds X in Tarski-Class X

proof end;

theorem Th3: :: CLASSES1:3

for X, Y, Z being set st Y in Tarski-Class X & Z c= Y holds
Z in Tarski-Class X

proof end;

theorem Th4: :: CLASSES1:4

for X, Y being set st Y in Tarski-Class X holds
bool Y in Tarski-Class X

proof end;

theorem Th5: :: CLASSES1:5

for X, Y being set holds
( not Y c= Tarski-Class X or Y, Tarski-Class X are_equipotent or Y in Tarski-Class X )

proof end;

theorem :: CLASSES1:6

for X, Y being set st Y c= Tarski-Class X & card Y in card (Tarski-Class X) holds
Y in Tarski-Class X

proof end;

definition

let X be set ;
let A be Ordinal;

func Tarski-Class (X,A) -> set means :Def5: :: CLASSES1:def 5
ex L being Sequence st
( it = last L & dom L = succ A & L . 0 = {X} & ( for C being Ordinal st succ C in succ A holds
L . (succ C) = ( { u where u is Element of Tarski-Class X : ex v being Element of Tarski-Class X st
( v in L . C & u c= v ) } \/ { (bool v) where v is Element of Tarski-Class X : v in L . C } ) \/ ((bool (L . C)) /\ (Tarski-Class X)) ) & ( for C being Ordinal st C in succ A & C <> 0 & C is limit_ordinal holds
L . C = (union (rng (L | C))) /\ (Tarski-Class X) ) );

correctness

proof end;

end;

:: deftheorem Def5 defines Tarski-Class CLASSES1:def 5 :

Lm1: now :: thesis:

let X be set ; :: thesis:
deffunc H₁( Ordinal) -> set = Tarski-Class (X,$1);
deffunc H₂( Ordinal, set ) -> set = ( { u where u is Element of Tarski-Class X : ex v being Element of Tarski-Class X st
( v in $2 & u c= v ) } \/ { (bool v) where v is Element of Tarski-Class X : v in $2 } ) \/ ((bool $2) /\ (Tarski-Class X));
deffunc H₃( Ordinal, Sequence) -> set = (union (rng $2)) /\ (Tarski-Class X);
A1: for A being Ordinal
for x being object holds
( x = H₁(A) iff ex L being Sequence st
( x = last L & dom L = succ A & L . 0 = {X} & ( for C being Ordinal st succ C in succ A holds
L . (succ C) = H₂(C,L . C) ) & ( for C being Ordinal st C in succ A & C <> 0 & C is limit_ordinal holds
L . C = H₃(C,L | C) ) ) ) by Def5;
thus H₁( 0 ) = {X} from ORDINAL2:sch 8(A1); :: thesis:
thus for A being Ordinal holds H₁( succ A) = H₂(A,H₁(A)) from ORDINAL2:sch 9(A1); :: thesis:
thus for A being Ordinal
for L being Sequence st A <> 0 & A is limit_ordinal & dom L = A & ( for B being Ordinal st B in A holds
L . B = Tarski-Class (X,B) ) holds
Tarski-Class (X,A) = (union (rng L)) /\ (Tarski-Class X) :: thesis:

proof

let A be Ordinal; :: thesis:
let L be Sequence; :: thesis:
assume that
A2: ( A <> 0 & A is limit_ordinal ) and
A3: dom L = A and
A4: for B being Ordinal st B in A holds
L . B = H₁(B) ; :: thesis:
thus H₁(A) = H₃(A,L) from ORDINAL2:sch 10(A1, A2, A3, A4); :: thesis:

end;

definition

let X be set ;
let A be Ordinal;
:: original: Tarski-Class
redefine func Tarski-Class (X,A) -> Subset of (Tarski-Class X);
coherence

proof end;

end;

theorem :: CLASSES1:7

for X being set holds Tarski-Class (X,{}) = {X} by Lm1;

theorem :: CLASSES1:8

for X being set
for A being Ordinal holds Tarski-Class (X,(succ A)) = ( { u where u is Element of Tarski-Class X : ex v being Element of Tarski-Class X st
( v in Tarski-Class (X,A) & u c= v ) } \/ { (bool v) where v is Element of Tarski-Class X : v in Tarski-Class (X,A) } ) \/ ((bool (Tarski-Class (X,A))) /\ (Tarski-Class X)) by Lm1;

theorem Th9: :: CLASSES1:9

for X being set
for A being Ordinal st A <> {} & A is limit_ordinal holds
Tarski-Class (X,A) = { u where u is Element of Tarski-Class X : ex B being Ordinal st
( B in A & u in Tarski-Class (X,B) ) }

proof end;

theorem Th10: :: CLASSES1:10

for X, Y being set
for A being Ordinal holds
( Y in Tarski-Class (X,(succ A)) iff ( ( Y c= Tarski-Class (X,A) & Y in Tarski-Class X ) or ex Z being set st
( Z in Tarski-Class (X,A) & ( Y c= Z or Y = bool Z ) ) ) )

proof end;

theorem :: CLASSES1:11

for X, Y, Z being set
for A being Ordinal st Y c= Z & Z in Tarski-Class (X,A) holds
Y in Tarski-Class (X,(succ A)) by Th10;

theorem :: CLASSES1:12

for X, Y being set
for A being Ordinal st Y in Tarski-Class (X,A) holds
bool Y in Tarski-Class (X,(succ A)) by Th10;

theorem Th13: :: CLASSES1:13

for X, x being set
for A being Ordinal st A <> {} & A is limit_ordinal holds
( x in Tarski-Class (X,A) iff ex B being Ordinal st
( B in A & x in Tarski-Class (X,B) ) )

proof end;

theorem :: CLASSES1:14

for X, Y, Z being set
for A being Ordinal st A <> {} & A is limit_ordinal & Y in Tarski-Class (X,A) & ( Z c= Y or Z = bool Y ) holds
Z in Tarski-Class (X,A)

proof end;

theorem Th15: :: CLASSES1:15

for X being set
for A being Ordinal holds Tarski-Class (X,A) c= Tarski-Class (X,(succ A))

proof end;

theorem Th16: :: CLASSES1:16

for X being set
for A, B being Ordinal st A c= B holds
Tarski-Class (X,A) c= Tarski-Class (X,B)

proof end;

theorem Th17: :: CLASSES1:17

for X being set ex A being Ordinal st Tarski-Class (X,A) = Tarski-Class (X,(succ A))

proof end;

theorem Th18: :: CLASSES1:18

for X being set
for A being Ordinal st Tarski-Class (X,A) = Tarski-Class (X,(succ A)) holds
Tarski-Class (X,A) = Tarski-Class X

proof end;

theorem Th19: :: CLASSES1:19

for X being set ex A being Ordinal st Tarski-Class (X,A) = Tarski-Class X

proof end;

theorem :: CLASSES1:20

for X being set ex A being Ordinal st
( Tarski-Class (X,A) = Tarski-Class X & ( for B being Ordinal st B in A holds
Tarski-Class (X,B) <> Tarski-Class X ) )

proof end;

theorem :: CLASSES1:21

for X, Y being set st Y <> X & Y in Tarski-Class X holds
ex A being Ordinal st
( not Y in Tarski-Class (X,A) & Y in Tarski-Class (X,(succ A)) )

proof end;

theorem Th22: :: CLASSES1:22

for X being set st X is epsilon-transitive holds
for A being Ordinal st A <> {} holds
Tarski-Class (X,A) is epsilon-transitive

proof end;

theorem Th23: :: CLASSES1:23

for X being set st X is epsilon-transitive holds
Tarski-Class X is epsilon-transitive

proof end;

theorem Th24: :: CLASSES1:24

for X, Y being set st Y in Tarski-Class X holds
card Y in card (Tarski-Class X)

proof end;

theorem Th25: :: CLASSES1:25

for X, Y being set st Y in Tarski-Class X holds
not Y, Tarski-Class X are_equipotent

proof end;

theorem Th26: :: CLASSES1:26

for X, x, y being set holds
( ( x in Tarski-Class X implies {x} in Tarski-Class X ) & ( x in Tarski-Class X & y in Tarski-Class X implies {x,y} in Tarski-Class X ) )

proof end;

theorem Th27: :: CLASSES1:27

for X, x, y being set st x in Tarski-Class X & y in Tarski-Class X holds
[x,y] in Tarski-Class X

proof end;

theorem :: CLASSES1:28

for X, Y, Z being set st Y c= Tarski-Class X & Z c= Tarski-Class X holds
[:Y,Z:] c= Tarski-Class X

proof end;

definition

let A be Ordinal;

func Rank A -> set means :Def6: :: CLASSES1:def 6
ex L being Sequence st
( it = last L & dom L = succ A & L . 0 = {} & ( for C being Ordinal st succ C in succ A holds
L . (succ C) = bool (L . C) ) & ( for C being Ordinal st C in succ A & C <> 0 & C is limit_ordinal holds
L . C = union (rng (L | C)) ) );

correctness

proof end;

end;

:: deftheorem Def6 defines Rank CLASSES1:def 6 :

deffunc H₁( Ordinal) -> set = Rank $1;

Lm2: now :: thesis:

deffunc H₂( Ordinal, set ) -> Element of bool (bool $2) = bool $2;
deffunc H₃( Ordinal, Sequence) -> set = union (rng $2);
A1: for A being Ordinal
for x being object holds
( x = H₁(A) iff ex L being Sequence st
( x = last L & dom L = succ A & L . 0 = {} & ( for C being Ordinal st succ C in succ A holds
L . (succ C) = H₂(C,L . C) ) & ( for C being Ordinal st C in succ A & C <> 0 & C is limit_ordinal holds
L . C = H₃(C,L | C) ) ) ) by Def6;
thus H₁( 0 ) = {} from ORDINAL2:sch 8(A1); :: thesis:
thus for A being Ordinal holds H₁( succ A) = H₂(A,H₁(A)) from ORDINAL2:sch 9(A1); :: thesis:
thus for A being Ordinal
for L being Sequence st A <> 0 & A is limit_ordinal & dom L = A & ( for B being Ordinal st B in A holds
L . B = Rank B ) holds
Rank A = union (rng L) :: thesis:

proof

end;

theorem :: CLASSES1:29

Rank {} = {} by Lm2;

theorem :: CLASSES1:30

for A being Ordinal holds Rank (succ A) = bool (Rank A) by Lm2;

theorem Th31: :: CLASSES1:31

for A being Ordinal st A <> {} & A is limit_ordinal holds
for x being set holds
( x in Rank A iff ex B being Ordinal st
( B in A & x in Rank B ) )

proof end;

theorem Th32: :: CLASSES1:32

for X being set
for A being Ordinal holds
( X c= Rank A iff X in Rank (succ A) )

proof end;

registration

let A be Ordinal;

cluster Rank A -> epsilon-transitive ;

coherence

proof end;

end;

theorem :: CLASSES1:33

for A being Ordinal holds Rank A c= Rank (succ A)

proof end;

theorem Th34: :: CLASSES1:34

for A being Ordinal holds union (Rank A) c= Rank A

proof end;

theorem :: CLASSES1:35

for X being set
for A being Ordinal st X in Rank A holds
union X in Rank A

proof end;

theorem Th36: :: CLASSES1:36

for A, B being Ordinal holds
( A in B iff Rank A in Rank B )

proof end;

theorem Th37: :: CLASSES1:37

for A, B being Ordinal holds
( A c= B iff Rank A c= Rank B )

proof end;

theorem Th38: :: CLASSES1:38

for A being Ordinal holds A c= Rank A

proof end;

theorem :: CLASSES1:39

for A being Ordinal
for X being set st X in Rank A holds
( not X, Rank A are_equipotent & card X in card (Rank A) )

proof end;

theorem :: CLASSES1:40

for X being set
for A being Ordinal holds
( X c= Rank A iff bool X c= Rank (succ A) )

proof end;

theorem Th41: :: CLASSES1:41

for X, Y being set
for A being Ordinal st X c= Y & Y in Rank A holds
X in Rank A

proof end;

theorem Th42: :: CLASSES1:42

for X being set
for A being Ordinal holds
( X in Rank A iff bool X in Rank (succ A) )

proof end;

theorem Th43: :: CLASSES1:43

for x being set
for A being Ordinal holds
( x in Rank A iff {x} in Rank (succ A) )

proof end;

theorem Th44: :: CLASSES1:44

for x, y being set
for A being Ordinal holds
( ( x in Rank A & y in Rank A ) iff {x,y} in Rank (succ A) )

proof end;

theorem :: CLASSES1:45

for x, y being set
for A being Ordinal holds
( ( x in Rank A & y in Rank A ) iff [x,y] in Rank (succ (succ A)) )

proof end;

theorem Th46: :: CLASSES1:46

for X being set
for A being Ordinal st X is epsilon-transitive & (Rank A) /\ (Tarski-Class X) = (Rank (succ A)) /\ (Tarski-Class X) holds
Tarski-Class X c= Rank A

proof end;

theorem Th47: :: CLASSES1:47

for X being set st X is epsilon-transitive holds
ex A being Ordinal st Tarski-Class X c= Rank A

proof end;

theorem Th48: :: CLASSES1:48

for X being set st X is epsilon-transitive holds
union X c= X

proof end;

theorem Th49: :: CLASSES1:49

for X, Y being set st X is epsilon-transitive & Y is epsilon-transitive holds
X \/ Y is epsilon-transitive

proof end;

theorem :: CLASSES1:50

for X, Y being set st X is epsilon-transitive & Y is epsilon-transitive holds
X /\ Y is epsilon-transitive

proof end;

deffunc H₂( set , set ) -> set = union $2;

definition

let X be set ;

func the_transitive-closure_of X -> set means :Def7: :: CLASSES1:def 7
for x being object holds
( x in it iff ex f being Function ex n being Element of omega st
( x in f . n & dom f = omega & f . 0 = X & ( for k being Nat holds f . (succ k) = union (f . k) ) ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def7 defines the_transitive-closure_of CLASSES1:def 7 :

registration

let X be set ;

cluster the_transitive-closure_of X -> epsilon-transitive ;

coherence

proof end;

end;

theorem :: CLASSES1:51

for X being set holds the_transitive-closure_of X is epsilon-transitive ;

theorem Th52: :: CLASSES1:52

for X being set holds X c= the_transitive-closure_of X

proof end;

theorem Th53: :: CLASSES1:53

for X, Y being set st X c= Y & Y is epsilon-transitive holds
the_transitive-closure_of X c= Y

proof end;

theorem Th54: :: CLASSES1:54

for X, Y being set st ( for Z being set st X c= Z & Z is epsilon-transitive holds
Y c= Z ) & X c= Y & Y is epsilon-transitive holds
the_transitive-closure_of X = Y by Th53, Th52;

theorem Th55: :: CLASSES1:55

for X being set st X is epsilon-transitive holds
the_transitive-closure_of X = X

proof end;

theorem :: CLASSES1:56

the_transitive-closure_of {} = {} by Th55;

theorem :: CLASSES1:57

for A being Ordinal holds the_transitive-closure_of A = A by Th55;

theorem Th58: :: CLASSES1:58

for X, Y being set st X c= Y holds
the_transitive-closure_of X c= the_transitive-closure_of Y

proof end;

theorem :: CLASSES1:59

for X being set holds the_transitive-closure_of (the_transitive-closure_of X) = the_transitive-closure_of X by Th55;

theorem :: CLASSES1:60

for X, Y being set holds the_transitive-closure_of (X \/ Y) = (the_transitive-closure_of X) \/ (the_transitive-closure_of Y)

proof end;

theorem :: CLASSES1:61

for X, Y being set holds the_transitive-closure_of (X /\ Y) c= (the_transitive-closure_of X) /\ (the_transitive-closure_of Y)

proof end;

theorem Th62: :: CLASSES1:62

for X being set ex A being Ordinal st X c= Rank A

proof end;

definition

let x be object ;

func the_rank_of x -> Ordinal means :Def8: :: CLASSES1:def 8
( x in Rank (succ it) & ( for B being Ordinal st x in Rank (succ B) holds
it c= B ) );

existence

proof end;

uniqueness ;

end;

:: deftheorem Def8 defines the_rank_of CLASSES1:def 8 :

definition

let X be set ;

redefine func the_rank_of X means :Def9: :: CLASSES1:def 9
( X c= Rank it & ( for B being Ordinal st X c= Rank B holds
it c= B ) );

compatibility

proof end;

end;

:: deftheorem Def9 defines the_rank_of CLASSES1:def 9 :

theorem Th63: :: CLASSES1:63

for X being set holds the_rank_of (bool X) = succ (the_rank_of X)

proof end;

theorem :: CLASSES1:64

for A being Ordinal holds the_rank_of (Rank A) = A

proof end;

theorem Th65: :: CLASSES1:65

for X being set
for A being Ordinal holds
( X c= Rank A iff the_rank_of X c= A )

proof end;

theorem Th66: :: CLASSES1:66

for X being set
for A being Ordinal holds
( X in Rank A iff the_rank_of X in A )

proof end;

theorem :: CLASSES1:67

for X, Y being set st X c= Y holds
the_rank_of X c= the_rank_of Y

proof end;

theorem Th68: :: CLASSES1:68

for X, Y being set st X in Y holds
the_rank_of X in the_rank_of Y

proof end;

theorem Th69: :: CLASSES1:69

for X being set
for A being Ordinal holds
( the_rank_of X c= A iff for Y being set st Y in X holds
the_rank_of Y in A )

proof end;

theorem Th70: :: CLASSES1:70

for X being set
for A being Ordinal holds
( A c= the_rank_of X iff for B being Ordinal st B in A holds
ex Y being set st
( Y in X & B c= the_rank_of Y ) )

proof end;

theorem :: CLASSES1:71

for X being set holds
( the_rank_of X = {} iff X = {} )

proof end;

theorem Th72: :: CLASSES1:72

for X being set
for A being Ordinal st the_rank_of X = succ A holds
ex Y being set st
( Y in X & the_rank_of Y = A )

proof end;

theorem :: CLASSES1:73

for A being Ordinal holds the_rank_of A = A

proof end;

theorem :: CLASSES1:74

for X being set holds
( the_rank_of (Tarski-Class X) <> {} & the_rank_of (Tarski-Class X) is limit_ordinal )

proof end;

scheme :: CLASSES1:sch 1
NonUniqFuncEx{ F₁() -> set , P₁[ object , object ] } :

ex f being Function st
( dom f = F₁() & ( for e being object st e in F₁() holds
P₁[e,f . e] ) )

provided

A1: for e being object st e in F₁() holds
ex u being object st P₁[e,u]

proof end;

:: from RFINSEQ, 2008.10.10, A.T.

definition

let F, G be Relation;

pred F,G are_fiberwise_equipotent means :: CLASSES1:def 10
for x being object holds card (Coim (F,x)) = card (Coim (G,x));

reflexivity ;
symmetry ;

end;

:: deftheorem defines are_fiberwise_equipotent CLASSES1:def 10 :

Lm3: for F being Function
for x being object st not x in rng F holds
Coim (F,x) = {}

proof end;

theorem Th75: :: CLASSES1:75

for F, G being Function st F,G are_fiberwise_equipotent holds
rng F = rng G

proof end;

theorem :: CLASSES1:76

for F, G, H being Function st F,G are_fiberwise_equipotent & F,H are_fiberwise_equipotent holds
G,H are_fiberwise_equipotent

proof end;

theorem Th77: :: CLASSES1:77

for F, G being Function holds
( F,G are_fiberwise_equipotent iff ex H being Function st
( dom H = dom F & rng H = dom G & H is one-to-one & F = G * H ) )

proof end;

theorem Th78: :: CLASSES1:78

for F, G being Function holds
( F,G are_fiberwise_equipotent iff for X being set holds card (F " X) = card (G " X) )

proof end;

theorem :: CLASSES1:79

for D being non empty set
for F, G being Function st rng F c= D & rng G c= D & ( for d being Element of D holds card (Coim (F,d)) = card (Coim (G,d)) ) holds
F,G are_fiberwise_equipotent

proof end;

theorem Th80: :: CLASSES1:80

for F, G being Function st dom F = dom G holds
( F,G are_fiberwise_equipotent iff ex P being Permutation of (dom F) st F = G * P )

proof end;

theorem :: CLASSES1:81

for F, G being Function st F,G are_fiberwise_equipotent holds
card (dom F) = card (dom G)

proof end;

:: from CIRCCOMB, 2009.01.26, A.T.

theorem :: CLASSES1:82

for f being set
for p being Relation
for x being set st x in rng p holds
the_rank_of x in the_rank_of [p,f]

proof end;

:: from BAGORDER, 2011.07.24, A.T.

theorem :: CLASSES1:83

for f, g, h being Function st dom f = dom g & rng f c= dom h & rng g c= dom h & f,g are_fiberwise_equipotent holds
h * f,h * g are_fiberwise_equipotent

proof end;

scheme :: CLASSES1:sch 2
LambdaAB{ F₁() -> set , F₂() -> set , F₃( Element of F₂()) -> set } :

ex f being Function st
( dom f = F₁() & ( for x being Element of F₂() st x in F₁() holds
f . x = F₃(x) ) )

proof end;

theorem :: CLASSES1:84

for x, y being set holds
( the_rank_of x in the_rank_of [x,y] & the_rank_of y in the_rank_of [x,y] )

proof end;