:: Domains and Their Cartesian Products
:: by Andrzej Trybulec
::
:: Received April 3, 1989
:: Copyright (c) 1990-2021 Association of Mizar Users

:: Domains and their elements

theorem :: DOMAIN_1:1

for a being set
for X1, X2 being non empty set st a in [:X1,X2:] holds
ex x1 being Element of X1 ex x2 being Element of X2 st a = [x1,x2]

proof end;

theorem :: DOMAIN_1:2

for X1, X2 being non empty set
for x, y being Element of [:X1,X2:] st x `1 = y `1 & x `2 = y `2 holds
x = y

proof end;

definition

let X1, X2 be non empty set ;
let x1 be Element of X1;
let x2 be Element of X2;
:: original: [
redefine func [x1,x2] -> Element of [:X1,X2:];
coherence
[x1,x2] is Element of [:X1,X2:] by ZFMISC_1:87;

end;

definition

let X1, X2 be non empty set ;
let x be Element of [:X1,X2:];
:: original: `1
redefine func x `1 -> Element of X1;
coherence
x `1 is Element of X1 by MCART_1:10;
:: original: `2
redefine func x `2 -> Element of X2;
coherence
x `2 is Element of X2 by MCART_1:10;

end;

:: Cartesian products of three sets

theorem Th3: :: DOMAIN_1:3

for a being set
for X1, X2, X3 being non empty set holds
( a in [:X1,X2,X3:] iff ex x1 being Element of X1 ex x2 being Element of X2 ex x3 being Element of X3 st a = [x1,x2,x3] )

proof end;

theorem Th4: :: DOMAIN_1:4

for D, X1, X2, X3 being non empty set st ( for a being set holds
( a in D iff ex x1 being Element of X1 ex x2 being Element of X2 ex x3 being Element of X3 st a = [x1,x2,x3] ) ) holds
D = [:X1,X2,X3:]

proof end;

theorem :: DOMAIN_1:5

for D, X1, X2, X3 being non empty set holds
( D = [:X1,X2,X3:] iff for a being set holds
( a in D iff ex x1 being Element of X1 ex x2 being Element of X2 ex x3 being Element of X3 st a = [x1,x2,x3] ) ) by Th3, Th4;

definition

let X1, X2, X3 be non empty set ;
let x1 be Element of X1;
let x2 be Element of X2;
let x3 be Element of X3;
:: original: [
redefine func [x1,x2,x3] -> Element of [:X1,X2,X3:];
coherence
[x1,x2,x3] is Element of [:X1,X2,X3:] by MCART_1:69;

end;

theorem :: DOMAIN_1:6

for a being set
for X1, X2, X3 being non empty set
for x being Element of [:X1,X2,X3:] holds
( a = x `1_3 iff for x1 being Element of X1
for x2 being Element of X2
for x3 being Element of X3 st x = [x1,x2,x3] holds
a = x1 ) by MCART_1:65;

theorem :: DOMAIN_1:7

for b being set
for X1, X2, X3 being non empty set
for x being Element of [:X1,X2,X3:] holds
( b = x `2_3 iff for x1 being Element of X1
for x2 being Element of X2
for x3 being Element of X3 st x = [x1,x2,x3] holds
b = x2 ) by MCART_1:66;

theorem :: DOMAIN_1:8

for c being set
for X1, X2, X3 being non empty set
for x being Element of [:X1,X2,X3:] holds
( c = x `3_3 iff for x1 being Element of X1
for x2 being Element of X2
for x3 being Element of X3 st x = [x1,x2,x3] holds
c = x3 ) by MCART_1:67;

AA: for X1, X2, X3 being non empty set
for x being Element of [:X1,X2,X3:] holds x = [(x `1_3),(x `2_3),(x `3_3)]
;

theorem :: DOMAIN_1:9

for X1, X2, X3 being non empty set
for x, y being Element of [:X1,X2,X3:] st x `1_3 = y `1_3 & x `2_3 = y `2_3 & x `3_3 = y `3_3 holds
x = y

proof end;

theorem Th10: :: DOMAIN_1:10

for a being set
for X1, X2, X3, X4 being non empty set holds
( a in [:X1,X2,X3,X4:] iff ex x1 being Element of X1 ex x2 being Element of X2 ex x3 being Element of X3 ex x4 being Element of X4 st a = [x1,x2,x3,x4] )

proof end;

theorem Th11: :: DOMAIN_1:11

for D, X1, X2, X3, X4 being non empty set st ( for a being set holds
( a in D iff ex x1 being Element of X1 ex x2 being Element of X2 ex x3 being Element of X3 ex x4 being Element of X4 st a = [x1,x2,x3,x4] ) ) holds
D = [:X1,X2,X3,X4:]

proof end;

theorem :: DOMAIN_1:12

for D, X1, X2, X3, X4 being non empty set holds
( D = [:X1,X2,X3,X4:] iff for a being set holds
( a in D iff ex x1 being Element of X1 ex x2 being Element of X2 ex x3 being Element of X3 ex x4 being Element of X4 st a = [x1,x2,x3,x4] ) ) by Th10, Th11;

definition

let X1, X2, X3, X4 be non empty set ;
let x1 be Element of X1;
let x2 be Element of X2;
let x3 be Element of X3;
let x4 be Element of X4;
:: original: [
redefine func [x1,x2,x3,x4] -> Element of [:X1,X2,X3,X4:];
coherence
[x1,x2,x3,x4] is Element of [:X1,X2,X3,X4:] by MCART_1:80;

end;

theorem :: DOMAIN_1:13

for a being set
for X1, X2, X3, X4 being non empty set
for x being Element of [:X1,X2,X3,X4:] holds
( a = x `1_4 iff for x1 being Element of X1
for x2 being Element of X2
for x3 being Element of X3
for x4 being Element of X4 st x = [x1,x2,x3,x4] holds
a = x1 ) by MCART_1:75;

theorem :: DOMAIN_1:14

for b being set
for X1, X2, X3, X4 being non empty set
for x being Element of [:X1,X2,X3,X4:] holds
( b = x `2_4 iff for x1 being Element of X1
for x2 being Element of X2
for x3 being Element of X3
for x4 being Element of X4 st x = [x1,x2,x3,x4] holds
b = x2 ) by MCART_1:76;

theorem :: DOMAIN_1:15

for c being set
for X1, X2, X3, X4 being non empty set
for x being Element of [:X1,X2,X3,X4:] holds
( c = x `3_4 iff for x1 being Element of X1
for x2 being Element of X2
for x3 being Element of X3
for x4 being Element of X4 st x = [x1,x2,x3,x4] holds
c = x3 ) by MCART_1:77;

theorem :: DOMAIN_1:16

for d being set
for X1, X2, X3, X4 being non empty set
for x being Element of [:X1,X2,X3,X4:] holds
( d = x `4_4 iff for x1 being Element of X1
for x2 being Element of X2
for x3 being Element of X3
for x4 being Element of X4 st x = [x1,x2,x3,x4] holds
d = x4 ) by MCART_1:78;

AB: for X1, X2, X3, X4 being non empty set
for x being Element of [:X1,X2,X3,X4:] holds x = [(x `1_4),(x `2_4),(x `3_4),(x `4_4)]
;

theorem :: DOMAIN_1:17

for X1, X2, X3, X4 being non empty set
for x, y being Element of [:X1,X2,X3,X4:] st x `1_4 = y `1_4 & x `2_4 = y `2_4 & x `3_4 = y `3_4 & x `4_4 = y `4_4 holds
x = y

proof end;

scheme :: DOMAIN_1:sch 1
Fraenkel1{ P₁[ object ] } :

for X1 being non empty set holds { x1 where x1 is Element of X1 : P₁[x1] } is Subset of X1

proof end;

scheme :: DOMAIN_1:sch 2
Fraenkel2{ P₁[ object , object ] } :

for X1, X2 being non empty set holds { [x1,x2] where x1 is Element of X1, x2 is Element of X2 : P₁[x1,x2] } is Subset of [:X1,X2:]

proof end;

scheme :: DOMAIN_1:sch 3
Fraenkel3{ P₁[ object , object , object ] } :

for X1, X2, X3 being non empty set holds { [x1,x2,x3] where x1 is Element of X1, x2 is Element of X2, x3 is Element of X3 : P₁[x1,x2,x3] } is Subset of [:X1,X2,X3:]

proof end;

scheme :: DOMAIN_1:sch 4
Fraenkel4{ P₁[ object , object , object , object ] } :

for X1, X2, X3, X4 being non empty set holds { [x1,x2,x3,x4] where x1 is Element of X1, x2 is Element of X2, x3 is Element of X3, x4 is Element of X4 : P₁[x1,x2,x3,x4] } is Subset of [:X1,X2,X3,X4:]

proof end;

scheme :: DOMAIN_1:sch 5
Fraenkel5{ P₁[ object ], P₂[ object ] } :

for X1 being non empty set st ( for x1 being Element of X1 st P₁[x1] holds
P₂[x1] ) holds
{ y1 where y1 is Element of X1 : P₁[y1] } c= { z1 where z1 is Element of X1 : P₂[z1] }

proof end;

scheme :: DOMAIN_1:sch 6
Fraenkel6{ P₁[ object ], P₂[ object ] } :

for X1 being non empty set st ( for x1 being Element of X1 holds
( P₁[x1] iff P₂[x1] ) ) holds
{ y1 where y1 is Element of X1 : P₁[y1] } = { z1 where z1 is Element of X1 : P₂[z1] }

proof end;

scheme :: DOMAIN_1:sch 7
SubsetD{ F₁() -> non empty set , P₁[ object ] } :

{ d where d is Element of F₁() : P₁[d] } is Subset of F₁()

proof end;

theorem :: DOMAIN_1:18

for X1 being non empty set holds X1 = { x1 where x1 is Element of X1 : verum }

proof end;

theorem :: DOMAIN_1:19

for X1, X2 being non empty set holds [:X1,X2:] = { [x1,x2] where x1 is Element of X1, x2 is Element of X2 : verum }

proof end;

theorem :: DOMAIN_1:20

for X1, X2, X3 being non empty set holds [:X1,X2,X3:] = { [x1,x2,x3] where x1 is Element of X1, x2 is Element of X2, x3 is Element of X3 : verum }

proof end;

theorem :: DOMAIN_1:21

for X1, X2, X3, X4 being non empty set holds [:X1,X2,X3,X4:] = { [x1,x2,x3,x4] where x1 is Element of X1, x2 is Element of X2, x3 is Element of X3, x4 is Element of X4 : verum }

proof end;

theorem :: DOMAIN_1:22

for X1 being non empty set
for A1 being Subset of X1 holds A1 = { x1 where x1 is Element of X1 : x1 in A1 }

proof end;

theorem :: DOMAIN_1:23

for X1, X2 being non empty set
for A1 being Subset of X1
for A2 being Subset of X2 holds [:A1,A2:] = { [x1,x2] where x1 is Element of X1, x2 is Element of X2 : ( x1 in A1 & x2 in A2 ) }

proof end;

theorem :: DOMAIN_1:24

for X1, X2, X3 being non empty set
for A1 being Subset of X1
for A2 being Subset of X2
for A3 being Subset of X3 holds [:A1,A2,A3:] = { [x1,x2,x3] where x1 is Element of X1, x2 is Element of X2, x3 is Element of X3 : ( x1 in A1 & x2 in A2 & x3 in A3 ) }

proof end;

theorem :: DOMAIN_1:25

for X1, X2, X3, X4 being non empty set
for A1 being Subset of X1
for A2 being Subset of X2
for A3 being Subset of X3
for A4 being Subset of X4 holds [:A1,A2,A3,A4:] = { [x1,x2,x3,x4] where x1 is Element of X1, x2 is Element of X2, x3 is Element of X3, x4 is Element of X4 : ( x1 in A1 & x2 in A2 & x3 in A3 & x4 in A4 ) }

proof end;

theorem :: DOMAIN_1:26

for X1 being non empty set holds {} X1 = { x1 where x1 is Element of X1 : contradiction }

proof end;

theorem :: DOMAIN_1:27

for X1 being non empty set
for A1 being Subset of X1 holds A1 ` = { x1 where x1 is Element of X1 : not x1 in A1 }

proof end;

theorem :: DOMAIN_1:28

for X1 being non empty set
for A1, B1 being Subset of X1 holds A1 /\ B1 = { x1 where x1 is Element of X1 : ( x1 in A1 & x1 in B1 ) }

proof end;

theorem :: DOMAIN_1:29

for X1 being non empty set
for A1, B1 being Subset of X1 holds A1 \/ B1 = { x1 where x1 is Element of X1 : ( x1 in A1 or x1 in B1 ) }

proof end;

theorem :: DOMAIN_1:30

for X1 being non empty set
for A1, B1 being Subset of X1 holds A1 \ B1 = { x1 where x1 is Element of X1 : ( x1 in A1 & not x1 in B1 ) }

proof end;

theorem Th31: :: DOMAIN_1:31

for X1 being non empty set
for A1, B1 being Subset of X1 holds A1 \+\ B1 = { x1 where x1 is Element of X1 : ( ( x1 in A1 & not x1 in B1 ) or ( not x1 in A1 & x1 in B1 ) ) }

proof end;

theorem Th32: :: DOMAIN_1:32

for X1 being non empty set
for A1, B1 being Subset of X1 holds A1 \+\ B1 = { x1 where x1 is Element of X1 : ( not x1 in A1 iff x1 in B1 ) }

proof end;

theorem :: DOMAIN_1:33

for X1 being non empty set
for A1, B1 being Subset of X1 holds A1 \+\ B1 = { x1 where x1 is Element of X1 : ( x1 in A1 iff not x1 in B1 ) }

proof end;

theorem :: DOMAIN_1:34

for X1 being non empty set
for A1, B1 being Subset of X1 holds A1 \+\ B1 = { x1 where x1 is Element of X1 : ( ( x1 in A1 & not x1 in B1 ) or ( x1 in B1 & not x1 in A1 ) ) }

proof end;

definition

let D be non empty set ;
let x1 be Element of D;
:: original: {
redefine func {x1} -> Subset of D;
coherence
{x1} is Subset of D by SUBSET_1:33;
let x2 be Element of D;
:: original: {
redefine func {x1,x2} -> Subset of D;
coherence
{x1,x2} is Subset of D by SUBSET_1:34;
let x3 be Element of D;
:: original: {
redefine func {x1,x2,x3} -> Subset of D;
coherence
{x1,x2,x3} is Subset of D by SUBSET_1:35;
let x4 be Element of D;
:: original: {
redefine func {x1,x2,x3,x4} -> Subset of D;
coherence
{x1,x2,x3,x4} is Subset of D by SUBSET_1:36;
let x5 be Element of D;
:: original: {
redefine func {x1,x2,x3,x4,x5} -> Subset of D;
coherence
{x1,x2,x3,x4,x5} is Subset of D by SUBSET_1:37;
let x6 be Element of D;
:: original: {
redefine func {x1,x2,x3,x4,x5,x6} -> Subset of D;
coherence
{x1,x2,x3,x4,x5,x6} is Subset of D by SUBSET_1:38;
let x7 be Element of D;
:: original: {
redefine func {x1,x2,x3,x4,x5,x6,x7} -> Subset of D;
coherence
{x1,x2,x3,x4,x5,x6,x7} is Subset of D by SUBSET_1:39;
let x8 be Element of D;
:: original: {
redefine func {x1,x2,x3,x4,x5,x6,x7,x8} -> Subset of D;
coherence
{x1,x2,x3,x4,x5,x6,x7,x8} is Subset of D by SUBSET_1:40;
let x9 be Element of D;
:: original: {
redefine func {x1,x2,x3,x4,x5,x6,x7,x8,x9} -> Subset of D;
coherence
{x1,x2,x3,x4,x5,x6,x7,x8,x9} is Subset of D by SUBSET_1:52;
let x10 be Element of D;
:: original: {
redefine func {x1,x2,x3,x4,x5,x6,x7,x8,x9,x10} -> Subset of D;
coherence
{x1,x2,x3,x4,x5,x6,x7,x8,x9,x10} is Subset of D by SUBSET_1:53;

end;

:: from COMPLSP1

scheme :: DOMAIN_1:sch 8
SubsetFD{ F₁() -> non empty set , F₂() -> non empty set , F₃( object ) -> Element of F₂(), P₁[ object ] } :

{ F₃(x) where x is Element of F₁() : P₁[x] } is Subset of F₂()

proof end;

scheme :: DOMAIN_1:sch 9
SubsetFD2{ F₁() -> non empty set , F₂() -> non empty set , F₃() -> non empty set , F₄( object , object ) -> Element of F₃(), P₁[ object , object ] } :

{ F₄(x,y) where x is Element of F₁(), y is Element of F₂() : P₁[x,y] } is Subset of F₃()

proof end;

definition

let D1, D2, D3 be non empty set ;
let x be Element of [:[:D1,D2:],D3:];
:: original: `11
redefine func x `11 -> Element of D1;
coherence
x `11 is Element of D1

proof end;

:: original: `12
redefine func x `12 -> Element of D2;
coherence
x `12 is Element of D2

proof end;

end;

definition

let D1, D2, D3 be non empty set ;
let x be Element of [:D1,[:D2,D3:]:];
:: original: `21
redefine func x `21 -> Element of D2;
coherence
x `21 is Element of D2

proof end;

:: original: `22
redefine func x `22 -> Element of D3;
coherence
x `22 is Element of D3

proof end;

end;

:: from JORDAN_A, 2005.10.12 , A.T.

scheme :: DOMAIN_1:sch 10
AndScheme{ F₁() -> non empty set , P₁[ object ], P₂[ object ] } :

{ a where a is Element of F₁() : ( P₁[a] & P₂[a] ) } = { a1 where a1 is Element of F₁() : P₁[a1] } /\ { a2 where a2 is Element of F₁() : P₂[a2] }

proof end;

:: from HAHNBAN, 2011.04.26, A.T.

registration

let A be non empty set ;

cluster non empty c=-linear for Element of bool A;

existence
ex b₁ being Subset of A st
( b₁ is c=-linear & not b₁ is empty )

proof end;

end;