:: Definitions and Basic Properties of Boolean & Union of Many Sorted Sets
:: by Artur Korni{\l}owicz
::
:: Copyright (c) 1995-2021 Association of Mizar Users

definition
let I be set ;
let A be ManySortedSet of I;
func bool A -> ManySortedSet of I means :Def1: :: MBOOLEAN:def 1
for i being object st i in I holds
it . i = bool (A . i);
existence
ex b1 being ManySortedSet of I st
for i being object st i in I holds
b1 . i = bool (A . i)
proof end;
uniqueness
for b1, b2 being ManySortedSet of I st ( for i being object st i in I holds
b1 . i = bool (A . i) ) & ( for i being object st i in I holds
b2 . i = bool (A . i) ) holds
b1 = b2
proof end;
end;

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

registration
let I be set ;
let A be ManySortedSet of I;
cluster bool A -> V2() ;
coherence
proof end;
end;

Lm1: for i, I, X being set
for M being ManySortedSet of I st i in I holds
dom (M +* (i .--> X)) = I

proof end;

Lm2: for I being set
for A, B being ManySortedSet of I
for i being set st i in I holds
(bool (A (\/) B)) . i = bool ((A . i) \/ (B . i))

proof end;

Lm3: for I being set
for A, B being ManySortedSet of I
for i being set st i in I holds
(bool (A (/\) B)) . i = bool ((A . i) /\ (B . i))

proof end;

Lm4: for I being set
for A, B being ManySortedSet of I
for i being set st i in I holds
(bool (A (\) B)) . i = bool ((A . i) \ (B . i))

proof end;

Lm5: for I being set
for A, B being ManySortedSet of I
for i being set st i in I holds
(bool (A (\+\) B)) . i = bool ((A . i) \+\ (B . i))

proof end;

theorem Th1: :: MBOOLEAN:1
for I being set
for X, Y being ManySortedSet of I holds
( X = bool Y iff for A being ManySortedSet of I holds
( A in X iff A c= Y ) )
proof end;

theorem :: MBOOLEAN:2
for I being set holds bool () = I -->
proof end;

theorem :: MBOOLEAN:3
for I, x being set holds bool (I --> x) = I --> (bool x)
proof end;

theorem :: MBOOLEAN:4
for x being object
for I being set holds bool (I --> {x}) = I --> {{},{x}}
proof end;

theorem :: MBOOLEAN:5
for I being set
for A being ManySortedSet of I holds EmptyMS I c= A
proof end;

theorem :: MBOOLEAN:6
for I being set
for A, B being ManySortedSet of I st A c= B holds
bool A c= bool B
proof end;

theorem :: MBOOLEAN:7
for I being set
for A, B being ManySortedSet of I holds (bool A) (\/) (bool B) c= bool (A (\/) B)
proof end;

theorem :: MBOOLEAN:8
for I being set
for A, B being ManySortedSet of I st (bool A) (\/) (bool B) = bool (A (\/) B) holds
for i being set st i in I holds
A . i,B . i are_c=-comparable
proof end;

theorem :: MBOOLEAN:9
for I being set
for A, B being ManySortedSet of I holds bool (A (/\) B) = (bool A) (/\) (bool B)
proof end;

theorem :: MBOOLEAN:10
for I being set
for A, B being ManySortedSet of I holds bool (A (\) B) c= () (\/) ((bool A) (\) (bool B))
proof end;

theorem :: MBOOLEAN:11
for I being set
for A, B, X being ManySortedSet of I holds
( X c= A (\) B iff ( X c= A & X misses B ) )
proof end;

theorem :: MBOOLEAN:12
for I being set
for A, B being ManySortedSet of I holds (bool (A (\) B)) (\/) (bool (B (\) A)) c= bool (A (\+\) B)
proof end;

theorem :: MBOOLEAN:13
for I being set
for A, B, X being ManySortedSet of I holds
( X c= A (\+\) B iff ( X c= A (\/) B & X misses A (/\) B ) )
proof end;

theorem :: MBOOLEAN:14
for I being set
for A, X, Y being ManySortedSet of I st ( X c= A or Y c= A ) holds
X (/\) Y c= A
proof end;

theorem :: MBOOLEAN:15
for I being set
for A, X, Y being ManySortedSet of I st X c= A holds
X (\) Y c= A
proof end;

theorem :: MBOOLEAN:16
for I being set
for A, X, Y being ManySortedSet of I st X c= A & Y c= A holds
X (\+\) Y c= A
proof end;

theorem :: MBOOLEAN:17
for I being set
for X, Y being ManySortedSet of I holds [|X,Y|] c= bool (bool (X (\/) Y))
proof end;

theorem :: MBOOLEAN:18
for I being set
for A, X being ManySortedSet of I holds
( X c= A iff X in bool A )
proof end;

theorem :: MBOOLEAN:19
for I being set
for A, B being ManySortedSet of I holds (Funcs) (A,B) c= bool [|A,B|]
proof end;

definition
let I be set ;
let A be ManySortedSet of I;
func union A -> ManySortedSet of I means :Def2: :: MBOOLEAN:def 2
for i being object st i in I holds
it . i = union (A . i);
existence
ex b1 being ManySortedSet of I st
for i being object st i in I holds
b1 . i = union (A . i)
proof end;
uniqueness
for b1, b2 being ManySortedSet of I st ( for i being object st i in I holds
b1 . i = union (A . i) ) & ( for i being object st i in I holds
b2 . i = union (A . i) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def2 defines union MBOOLEAN:def 2 :
for I being set
for A, b3 being ManySortedSet of I holds
( b3 = union A iff for i being object st i in I holds
b3 . i = union (A . i) );

registration
let I be set ;
cluster union () -> V3() ;
coherence
proof end;
end;

Lm6: for I being set
for A, B being ManySortedSet of I
for i being set st i in I holds
(union (A (\/) B)) . i = union ((A . i) \/ (B . i))

proof end;

Lm7: for I being set
for A, B being ManySortedSet of I
for i being set st i in I holds
(union (A (/\) B)) . i = union ((A . i) /\ (B . i))

proof end;

theorem :: MBOOLEAN:20
for I being set
for A, X being ManySortedSet of I holds
( A in union X iff ex Y being ManySortedSet of I st
( A in Y & Y in X ) )
proof end;

theorem :: MBOOLEAN:21
for I being set holds union () = EmptyMS I
proof end;

theorem :: MBOOLEAN:22
for I, x being set holds union (I --> x) = I --> ()
proof end;

theorem :: MBOOLEAN:23
for x being object
for I being set holds union (I --> {x}) = I --> x
proof end;

theorem :: MBOOLEAN:24
for x, y being object
for I being set holds union (I --> {{x},{y}}) = I --> {x,y}
proof end;

theorem :: MBOOLEAN:25
for I being set
for A, X being ManySortedSet of I st X in A holds
X c= union A
proof end;

theorem :: MBOOLEAN:26
for I being set
for A, B being ManySortedSet of I st A c= B holds
union A c= union B
proof end;

theorem :: MBOOLEAN:27
for I being set
for A, B being ManySortedSet of I holds union (A (\/) B) = () (\/) ()
proof end;

theorem :: MBOOLEAN:28
for I being set
for A, B being ManySortedSet of I holds union (A (/\) B) c= () (/\) ()
proof end;

theorem :: MBOOLEAN:29
for I being set
for A being ManySortedSet of I holds union (bool A) = A
proof end;

theorem :: MBOOLEAN:30
for I being set
for A being ManySortedSet of I holds A c= bool ()
proof end;

theorem :: MBOOLEAN:31
for I being set
for A, X, Y being ManySortedSet of I st union Y c= A & X in Y holds
X c= A
proof end;

theorem :: MBOOLEAN:32
for I being set
for Z being ManySortedSet of I
for A being V2() ManySortedSet of I st ( for X being ManySortedSet of I st X in A holds
X c= Z ) holds
union A c= Z
proof end;

theorem :: MBOOLEAN:33
for I being set
for B being ManySortedSet of I
for A being V2() ManySortedSet of I st ( for X being ManySortedSet of I st X in A holds
X (/\) B = EmptyMS I ) holds
() (/\) B = EmptyMS I
proof end;

theorem :: MBOOLEAN:34
for I being set
for A, B being ManySortedSet of I st A (\/) B is V2() & ( for X, Y being ManySortedSet of I st X <> Y & X in A (\/) B & Y in A (\/) B holds
X (/\) Y = EmptyMS I ) holds
union (A (/\) B) = () (/\) ()
proof end;

theorem :: MBOOLEAN:35
for I being set
for A, X being ManySortedSet of I
for B being V2() ManySortedSet of I st X c= union (A (\/) B) & ( for Y being ManySortedSet of I st Y in B holds
Y (/\) X = EmptyMS I ) holds
X c= union A
proof end;

theorem :: MBOOLEAN:36
for I being set
for A being V2() V26() ManySortedSet of I st ( for X, Y being ManySortedSet of I st X in A & Y in A & not X c= Y holds
Y c= X ) holds
union A in A
proof end;