:: Some Basic Properties of Many Sorted Sets
:: by Artur Korni{\l}owicz
::
:: Received September 29, 1995
:: Copyright (c) 1995-2021 Association of Mizar Users

theorem Th1: :: PZFMISC1:1

for i being object
for I being set
for X being object
for M being ManySortedSet of I st i in I holds
dom (M +* (i .--> X)) = I

proof end;

theorem :: PZFMISC1:2

for f being Function st f = {} holds
f is ManySortedSet of {} by PARTFUN1:def 2, RELAT_1:38, RELAT_1:def 18;

theorem :: PZFMISC1:3

for I being set st not I is empty holds
for X being ManySortedSet of I holds
( not X is V3() or not X is V2() )

proof end;

definition

let I be set ;
let A be ManySortedSet of I;

func {A} -> ManySortedSet of I means :Def1: :: PZFMISC1:def 1
for i being object st i in I holds
it . i = {(A . i)};

existence
ex b₁ being ManySortedSet of I st
for i being object st i in I holds
b₁ . i = {(A . i)}

proof end;

uniqueness
for b₁, b₂ being ManySortedSet of I st ( for i being object st i in I holds
b₁ . i = {(A . i)} ) & ( for i being object st i in I holds
b₂ . i = {(A . i)} ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def1 defines { PZFMISC1:def 1 :
for I being set
for A, b₃ being ManySortedSet of I holds
( b₃ = {A} iff for i being object st i in I holds
b₃ . i = {(A . i)} );

registration

let I be set ;
let A be ManySortedSet of I;

cluster {A} -> V2() V26() ;

coherence
( {A} is non-empty & {A} is finite-yielding )

proof end;

end;

definition

let I be set ;
let A, B be ManySortedSet of I;

func {A,B} -> ManySortedSet of I means :Def2: :: PZFMISC1:def 2
for i being object st i in I holds
it . i = {(A . i),(B . i)};

existence
ex b₁ being ManySortedSet of I st
for i being object st i in I holds
b₁ . i = {(A . i),(B . i)}

proof end;

uniqueness
for b₁, b₂ being ManySortedSet of I st ( for i being object st i in I holds
b₁ . i = {(A . i),(B . i)} ) & ( for i being object st i in I holds
b₂ . i = {(A . i),(B . i)} ) holds
b₁ = b₂

proof end;

commutativity
for b₁, A, B being ManySortedSet of I st ( for i being object st i in I holds
b₁ . i = {(A . i),(B . i)} ) holds
for i being object st i in I holds
b₁ . i = {(B . i),(A . i)} ;

end;

:: deftheorem Def2 defines { PZFMISC1:def 2 :
for I being set
for A, B, b₄ being ManySortedSet of I holds
( b₄ = {A,B} iff for i being object st i in I holds
b₄ . i = {(A . i),(B . i)} );

registration

let I be set ;
let A, B be ManySortedSet of I;

cluster {A,B} -> V2() V26() ;

coherence
( {A,B} is non-empty & {A,B} is finite-yielding )

proof end;

end;

theorem :: PZFMISC1:4

for I being set
for y, X being ManySortedSet of I holds
( X = {y} iff for x being ManySortedSet of I holds
( x in X iff x = y ) )

proof end;

theorem :: PZFMISC1:5

for I being set
for x1, x2, X being ManySortedSet of I st ( for x being ManySortedSet of I holds
( x in X iff ( x = x1 or x = x2 ) ) ) holds
X = {x1,x2}

proof end;

theorem :: PZFMISC1:6

for I being set
for x1, x2, X being ManySortedSet of I st X = {x1,x2} holds
for x being ManySortedSet of I st ( x = x1 or x = x2 ) holds
x in X

proof end;

theorem :: PZFMISC1:7

for I being set
for x, A being ManySortedSet of I st x in {A} holds
x = A

proof end;

theorem :: PZFMISC1:8

for I being set
for x being ManySortedSet of I holds x in {x}

proof end;

theorem :: PZFMISC1:9

for I being set
for x, A, B being ManySortedSet of I st ( x = A or x = B ) holds
x in {A,B}

proof end;

theorem :: PZFMISC1:10

for I being set
for A, B being ManySortedSet of I holds {A} (\/) {B} = {A,B}

proof end;

theorem :: PZFMISC1:11

for I being set
for x being ManySortedSet of I holds {x,x} = {x}

proof end;

theorem :: PZFMISC1:12

for I being set
for A, B being ManySortedSet of I st {A} c= {B} holds
A = B

proof end;

theorem :: PZFMISC1:13

for I being set
for x, y being ManySortedSet of I st {x} = {y} holds
x = y

proof end;

theorem :: PZFMISC1:14

for I being set
for x, A, B being ManySortedSet of I st {x} = {A,B} holds
( x = A & x = B )

proof end;

theorem :: PZFMISC1:15

for I being set
for x, A, B being ManySortedSet of I st {x} = {A,B} holds
A = B

proof end;

theorem :: PZFMISC1:16

for I being set
for x, y being ManySortedSet of I holds
( {x} c= {x,y} & {y} c= {x,y} )

proof end;

theorem :: PZFMISC1:17

for I being set
for x, y being ManySortedSet of I st {x} (\/) {y} = {x} holds
x = y

proof end;

theorem :: PZFMISC1:18

for I being set
for x, y being ManySortedSet of I holds {x} (\/) {x,y} = {x,y}

proof end;

theorem :: PZFMISC1:19

for I being set
for x, y being ManySortedSet of I st not I is empty & {x} (/\) {y} = EmptyMS I holds
x <> y

proof end;

theorem :: PZFMISC1:20

for I being set
for x, y being ManySortedSet of I st {x} (/\) {y} = {x} holds
x = y

proof end;

theorem :: PZFMISC1:21

for I being set
for x, y being ManySortedSet of I holds {x} (/\) {x,y} = {x}

proof end;

theorem :: PZFMISC1:22

for I being set
for x, y being ManySortedSet of I st not I is empty & {x} (\) {y} = {x} holds
x <> y

proof end;

theorem :: PZFMISC1:23

for I being set
for x, y being ManySortedSet of I st {x} (\) {y} = EmptyMS I holds
x = y

proof end;

theorem :: PZFMISC1:24

for I being set
for x, y being ManySortedSet of I holds {x} (\) {x,y} = EmptyMS I

proof end;

theorem :: PZFMISC1:25

for I being set
for x, y being ManySortedSet of I st {x} c= {y} holds
{x} = {y}

proof end;

theorem :: PZFMISC1:26

for I being set
for x, y, A being ManySortedSet of I st {x,y} c= {A} holds
( x = A & y = A )

proof end;

theorem :: PZFMISC1:27

for I being set
for x, y, A being ManySortedSet of I st {x,y} c= {A} holds
{x,y} = {A}

proof end;

theorem :: PZFMISC1:28

for I being set
for x being ManySortedSet of I holds bool {x} = {(EmptyMS I),{x}}

proof end;

theorem :: PZFMISC1:29

for I being set
for A being ManySortedSet of I holds {A} c= bool A

proof end;

theorem :: PZFMISC1:30

for I being set
for x being ManySortedSet of I holds union {x} = x

proof end;

theorem :: PZFMISC1:31

for I being set
for x, y being ManySortedSet of I holds union {{x},{y}} = {x,y}

proof end;

theorem :: PZFMISC1:32

for I being set
for A, B being ManySortedSet of I holds union {A,B} = A (\/) B

proof end;

theorem :: PZFMISC1:33

for I being set
for x, X being ManySortedSet of I holds
( {x} c= X iff x in X )

proof end;

theorem :: PZFMISC1:34

for I being set
for x1, x2, X being ManySortedSet of I holds
( {x1,x2} c= X iff ( x1 in X & x2 in X ) )

proof end;

theorem :: PZFMISC1:35

for I being set
for x1, x2, A being ManySortedSet of I st ( A = EmptyMS I or A = {x1} or A = {x2} or A = {x1,x2} ) holds
A c= {x1,x2}

proof end;

theorem :: PZFMISC1:36

for I being set
for x, A, B being ManySortedSet of I st ( x in A or x = B ) holds
x in A (\/) {B}

proof end;

theorem :: PZFMISC1:37

for I being set
for x, A, B being ManySortedSet of I holds
( A (\/) {x} c= B iff ( x in B & A c= B ) )

proof end;

theorem :: PZFMISC1:38

for I being set
for x, X being ManySortedSet of I st {x} (\/) X = X holds
x in X

proof end;

theorem :: PZFMISC1:39

for I being set
for x, X being ManySortedSet of I st x in X holds
{x} (\/) X = X

proof end;

theorem :: PZFMISC1:40

for I being set
for x, y, A being ManySortedSet of I holds
( {x,y} (\/) A = A iff ( x in A & y in A ) )

proof end;

theorem :: PZFMISC1:41

for I being set
for x, X being ManySortedSet of I st not I is empty holds
{x} (\/) X <> EmptyMS I

proof end;

theorem :: PZFMISC1:42

for I being set
for x, y, X being ManySortedSet of I st not I is empty holds
{x,y} (\/) X <> EmptyMS I

proof end;

theorem :: PZFMISC1:43

for I being set
for x, X being ManySortedSet of I st X (/\) {x} = {x} holds
x in X

proof end;

theorem :: PZFMISC1:44

for I being set
for x, X being ManySortedSet of I st x in X holds
X (/\) {x} = {x}

proof end;

theorem :: PZFMISC1:45

for I being set
for x, y, X being ManySortedSet of I holds
( ( x in X & y in X ) iff {x,y} (/\) X = {x,y} )

proof end;

theorem :: PZFMISC1:46

for I being set
for x, X being ManySortedSet of I st not I is empty & {x} (/\) X = EmptyMS I holds
not x in X

proof end;

theorem :: PZFMISC1:47

for I being set
for x, y, X being ManySortedSet of I st not I is empty & {x,y} (/\) X = EmptyMS I holds
( not x in X & not y in X )

proof end;

theorem :: PZFMISC1:48

for I being set
for x, y, X being ManySortedSet of I st y in X (\) {x} holds
y in X

proof end;

theorem :: PZFMISC1:49

for I being set
for x, y, X being ManySortedSet of I st not I is empty & y in X (\) {x} holds
y <> x

proof end;

theorem :: PZFMISC1:50

for I being set
for x, X being ManySortedSet of I st not I is empty & X (\) {x} = X holds
not x in X

proof end;

theorem :: PZFMISC1:51

for I being set
for x, X being ManySortedSet of I st not I is empty & {x} (\) X = {x} holds
not x in X

proof end;

theorem :: PZFMISC1:52

for I being set
for x, X being ManySortedSet of I holds
( {x} (\) X = EmptyMS I iff x in X )

proof end;

theorem :: PZFMISC1:53

for I being set
for x, y, X being ManySortedSet of I st not I is empty & {x,y} (\) X = {x} holds
not x in X

proof end;

theorem :: PZFMISC1:54

for I being set
for x, y, X being ManySortedSet of I st not I is empty & {x,y} (\) X = {x,y} holds
( not x in X & not y in X )

proof end;

theorem :: PZFMISC1:55

for I being set
for x, y, X being ManySortedSet of I holds
( {x,y} (\) X = EmptyMS I iff ( x in X & y in X ) )

proof end;

theorem :: PZFMISC1:56

for I being set
for x, y, X being ManySortedSet of I st ( X = EmptyMS I or X = {x} or X = {y} or X = {x,y} ) holds
X (\) {x,y} = EmptyMS I

proof end;

theorem :: PZFMISC1:57

for I being set
for X, Y being ManySortedSet of I st ( X = EmptyMS I or Y = EmptyMS I ) holds
[|X,Y|] = EmptyMS I

proof end;

theorem :: PZFMISC1:58

for I being set
for X, Y being ManySortedSet of I st X is V2() & Y is V2() & [|X,Y|] = [|Y,X|] holds
X = Y

proof end;

theorem :: PZFMISC1:59

for I being set
for X, Y being ManySortedSet of I st [|X,X|] = [|Y,Y|] holds
X = Y

proof end;

theorem :: PZFMISC1:60

for I being set
for X, Y, Z being ManySortedSet of I st Z is V2() & ( [|X,Z|] c= [|Y,Z|] or [|Z,X|] c= [|Z,Y|] ) holds
X c= Y

proof end;

theorem :: PZFMISC1:61

for I being set
for X, Y, Z being ManySortedSet of I st X c= Y holds
( [|X,Z|] c= [|Y,Z|] & [|Z,X|] c= [|Z,Y|] )

proof end;

theorem :: PZFMISC1:62

for I being set
for x1, x2, A, B being ManySortedSet of I st x1 c= A & x2 c= B holds
[|x1,x2|] c= [|A,B|]

proof end;

theorem :: PZFMISC1:63

for I being set
for X, Y, Z being ManySortedSet of I holds
( [|(X (\/) Y),Z|] = [|X,Z|] (\/) [|Y,Z|] & [|Z,(X (\/) Y)|] = [|Z,X|] (\/) [|Z,Y|] )

proof end;

theorem :: PZFMISC1:64

for I being set
for x1, x2, A, B being ManySortedSet of I holds [|(x1 (\/) x2),(A (\/) B)|] = (([|x1,A|] (\/) [|x1,B|]) (\/) [|x2,A|]) (\/) [|x2,B|]

proof end;

theorem :: PZFMISC1:65

for I being set
for X, Y, Z being ManySortedSet of I holds
( [|(X (/\) Y),Z|] = [|X,Z|] (/\) [|Y,Z|] & [|Z,(X (/\) Y)|] = [|Z,X|] (/\) [|Z,Y|] )

proof end;

theorem :: PZFMISC1:66

for I being set
for x1, x2, A, B being ManySortedSet of I holds [|(x1 (/\) x2),(A (/\) B)|] = [|x1,A|] (/\) [|x2,B|]

proof end;

theorem :: PZFMISC1:67

for I being set
for A, B, X, Y being ManySortedSet of I st A c= X & B c= Y holds
[|A,Y|] (/\) [|X,B|] = [|A,B|]

proof end;

theorem :: PZFMISC1:68

for I being set
for X, Y, Z being ManySortedSet of I holds
( [|(X (\) Y),Z|] = [|X,Z|] (\) [|Y,Z|] & [|Z,(X (\) Y)|] = [|Z,X|] (\) [|Z,Y|] )

proof end;

theorem :: PZFMISC1:69

for I being set
for x1, x2, A, B being ManySortedSet of I holds [|x1,x2|] (\) [|A,B|] = [|(x1 (\) A),x2|] (\/) [|x1,(x2 (\) B)|]

proof end;

theorem :: PZFMISC1:70

for I being set
for x1, x2, A, B being ManySortedSet of I st ( x1 (/\) x2 = EmptyMS I or A (/\) B = EmptyMS I ) holds
[|x1,A|] (/\) [|x2,B|] = EmptyMS I

proof end;

theorem :: PZFMISC1:71

for I being set
for x, X being ManySortedSet of I st X is V2() holds
( [|{x},X|] is V2() & [|X,{x}|] is V2() )

proof end;

theorem :: PZFMISC1:72

for I being set
for x, y, X being ManySortedSet of I holds
( [|{x,y},X|] = [|{x},X|] (\/) [|{y},X|] & [|X,{x,y}|] = [|X,{x}|] (\/) [|X,{y}|] )

proof end;

theorem :: PZFMISC1:73

for I being set
for x1, x2, A, B being ManySortedSet of I st x1 is V2() & A is V2() & [|x1,A|] = [|x2,B|] holds
( x1 = x2 & A = B )

proof end;

theorem :: PZFMISC1:74

for I being set
for X, Y being ManySortedSet of I st ( X c= [|X,Y|] or X c= [|Y,X|] ) holds
X = EmptyMS I

proof end;

theorem :: PZFMISC1:75

for I being set
for x, y, A, X, Y being ManySortedSet of I st A in [|x,y|] & A in [|X,Y|] holds
A in [|(x (/\) X),(y (/\) Y)|]

proof end;

theorem :: PZFMISC1:76

for I being set
for x, y, X, Y being ManySortedSet of I st [|x,X|] c= [|y,Y|] & [|x,X|] is V2() holds
( x c= y & X c= Y )

proof end;

theorem :: PZFMISC1:77

for I being set
for A, X being ManySortedSet of I st A c= X holds
[|A,A|] c= [|X,X|]

proof end;

theorem :: PZFMISC1:78

for I being set
for X, Y being ManySortedSet of I st X (/\) Y = EmptyMS I holds
[|X,Y|] (/\) [|Y,X|] = EmptyMS I

proof end;

theorem :: PZFMISC1:79

for I being set
for A, B, X, Y being ManySortedSet of I st A is V2() & ( [|A,B|] c= [|X,Y|] or [|B,A|] c= [|Y,X|] ) holds
B c= Y

proof end;

theorem :: PZFMISC1:80

for I being set
for x, y, A, B, X, Y being ManySortedSet of I st x c= [|A,B|] & y c= [|X,Y|] holds
x (\/) y c= [|(A (\/) X),(B (\/) Y)|]

proof end;

:: from AUTALG_1

definition

let I be set ;
let A, B be ManySortedSet of I;

pred A is_transformable_to B means :: PZFMISC1:def 3
for i being set st i in I & B . i = {} holds
A . i = {} ;

reflexivity
for A being ManySortedSet of I
for i being set st i in I & A . i = {} holds
A . i = {} ;

end;

:: deftheorem defines is_transformable_to PZFMISC1:def 3 :
for I being set
for A, B being ManySortedSet of I holds
( A is_transformable_to B iff for i being set st i in I & B . i = {} holds
A . i = {} );