:: Some Properties of Binary Relations
:: by Waldemar Korczy\'nski
::
:: Received January 17, 1992
:: Copyright (c) 1992-2021 Association of Mizar Users

Lm1: for X, Y being set st X <> {} & Y <> {} holds
( dom [:X,Y:] = X & rng [:X,Y:] = Y )
by RELAT_1:160;

theorem Th1: :: SYSREL:1

for X, Y being set
for R being Relation holds
( dom (R /\ [:X,Y:]) c= X & rng (R /\ [:X,Y:]) c= Y )

proof end;

theorem :: SYSREL:2

for X, Y being set
for R being Relation st X misses Y holds
dom (R /\ [:X,Y:]) misses rng (R /\ [:X,Y:])

proof end;

theorem Th3: :: SYSREL:3

for X, Y being set
for R being Relation st R c= [:X,Y:] holds
( dom R c= X & rng R c= Y )

proof end;

theorem :: SYSREL:4

for X, Y being set
for R being Relation st R c= [:X,Y:] holds
R ~ c= [:Y,X:]

proof end;

theorem :: SYSREL:5

for X, Y being set holds [:X,Y:] ~ = [:Y,X:]

proof end;

theorem Th6: :: SYSREL:6

for R, S, T being Relation holds (R \/ S) * T = (R * T) \/ (S * T)

proof end;

theorem :: SYSREL:7

for x, y being object
for X, Y being set
for R being Relation holds
( ( X misses Y & R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R & x in X implies ( not x in Y & not y in X & y in Y ) ) & ( X misses Y & R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R & y in Y implies ( not y in X & not x in Y & x in X ) ) & ( X misses Y & R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R & x in Y implies ( not x in X & not y in Y & y in X ) ) & ( X misses Y & R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R & y in X implies ( not x in X & not y in Y & x in Y ) ) )

proof end;

theorem Th8: :: SYSREL:8

for X, Y, Z being set
for R being Relation st R c= [:X,Y:] holds
( R | Z = R /\ [:Z,Y:] & Z |` R = R /\ [:X,Z:] )

proof end;

theorem :: SYSREL:9

for X, Y, Z, W being set
for R being Relation st R c= [:X,Y:] & X = Z \/ W holds
R = (R | Z) \/ (R | W)

proof end;

theorem :: SYSREL:10

for X, Y being set
for R being Relation st X misses Y & R c= [:X,Y:] \/ [:Y,X:] holds
R | X c= [:X,Y:]

proof end;

theorem :: SYSREL:11

for R, S being Relation st R c= S holds
R ~ c= S ~

proof end;

Lm2: for X being set holds id X c= [:X,X:]

proof end;

theorem Th12: :: SYSREL:12

for X being set holds (id X) * (id X) = id X

proof end;

theorem :: SYSREL:13

for x being object holds id {x} = {[x,x]}

proof end;

theorem Th14: :: SYSREL:14

for X, Y being set holds
( id (X \/ Y) = (id X) \/ (id Y) & id (X /\ Y) = (id X) /\ (id Y) & id (X \ Y) = (id X) \ (id Y) )

proof end;

theorem Th15: :: SYSREL:15

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

proof end;

theorem :: SYSREL:16

for X, Y being set holds (id (X \ Y)) \ (id X) = {} by Th15, XBOOLE_1:37;

theorem :: SYSREL:17

for R being Relation st R c= id (dom R) holds
R = id (dom R)

proof end;

theorem :: SYSREL:18

for X being set
for R being Relation st id X c= R \/ (R ~) holds
( id X c= R & id X c= R ~ )

proof end;

theorem :: SYSREL:19

for X being set
for R being Relation st id X c= R holds
id X c= R ~

proof end;

:: CLOSURE RELATION

theorem :: SYSREL:20

for X being set
for R being Relation st R c= [:X,X:] holds
( R \ (id (dom R)) = R \ (id X) & R \ (id (rng R)) = R \ (id X) )

proof end;

theorem :: SYSREL:21

for X being set
for R being Relation holds
( ( (id X) * (R \ (id X)) = {} implies dom (R \ (id X)) = (dom R) \ X ) & ( (R \ (id X)) * (id X) = {} implies rng (R \ (id X)) = (rng R) \ X ) )

proof end;

theorem Th22: :: SYSREL:22

for R being Relation holds
( ( R c= R * R & R * (R \ (id (rng R))) = {} implies id (rng R) c= R ) & ( R c= R * R & (R \ (id (dom R))) * R = {} implies id (dom R) c= R ) )

proof end;

theorem :: SYSREL:23

for R being Relation holds
( ( R c= R * R & R * (R \ (id (rng R))) = {} implies R /\ (id (rng R)) = id (rng R) ) & ( R c= R * R & (R \ (id (dom R))) * R = {} implies R /\ (id (dom R)) = id (dom R) ) ) by Th22, XBOOLE_1:28;

theorem :: SYSREL:24

for X being set
for R being Relation holds
( ( R * (R \ (id X)) = {} implies R * (R \ (id (rng R))) = {} ) & ( (R \ (id X)) * R = {} implies (R \ (id (dom R))) * R = {} ) )

proof end;

:: OPERATION CL

definition

let R be Relation;

func CL R -> Relation equals :: SYSREL:def 1
R /\ (id (dom R));

correctness
coherence
R /\ (id (dom R)) is Relation;
;

end;

:: deftheorem defines CL SYSREL:def 1 :
for R being Relation holds CL R = R /\ (id (dom R));

theorem Th25: :: SYSREL:25

for R being Relation
for x, y being object st [x,y] in CL R holds
( x in dom (CL R) & x = y )

proof end;

theorem Th26: :: SYSREL:26

for R being Relation holds dom (CL R) = rng (CL R)

proof end;

theorem Th27: :: SYSREL:27

for x being object
for R being Relation holds
( ( x in dom (CL R) implies ( x in dom R & [x,x] in R ) ) & ( x in dom R & [x,x] in R implies x in dom (CL R) ) & ( x in rng (CL R) implies ( x in dom R & [x,x] in R ) ) & ( x in dom R & [x,x] in R implies x in rng (CL R) ) & ( x in rng (CL R) implies ( x in rng R & [x,x] in R ) ) & ( x in rng R & [x,x] in R implies x in rng (CL R) ) & ( x in dom (CL R) implies ( x in rng R & [x,x] in R ) ) & ( x in rng R & [x,x] in R implies x in dom (CL R) ) )

proof end;

theorem Th28: :: SYSREL:28

for R being Relation holds CL R = id (dom (CL R))

proof end;

theorem Th29: :: SYSREL:29

for x, y being object
for R being Relation holds
( ( R * R = R & R * (R \ (CL R)) = {} & [x,y] in R & x <> y implies ( x in (dom R) \ (dom (CL R)) & y in dom (CL R) ) ) & ( R * R = R & (R \ (CL R)) * R = {} & [x,y] in R & x <> y implies ( y in (rng R) \ (dom (CL R)) & x in dom (CL R) ) ) )

proof end;

theorem :: SYSREL:30

for x, y being object
for R being Relation holds
( ( R * R = R & R * (R \ (id (dom R))) = {} & [x,y] in R & x <> y implies ( x in (dom R) \ (dom (CL R)) & y in dom (CL R) ) ) & ( R * R = R & (R \ (id (dom R))) * R = {} & [x,y] in R & x <> y implies ( y in (rng R) \ (dom (CL R)) & x in dom (CL R) ) ) )

proof end;

theorem :: SYSREL:31

for R being Relation holds
( ( R * R = R & R * (R \ (id (dom R))) = {} implies ( dom (CL R) = rng R & rng (CL R) = rng R ) ) & ( R * R = R & (R \ (id (dom R))) * R = {} implies ( dom (CL R) = dom R & rng (CL R) = dom R ) ) )

proof end;

theorem Th32: :: SYSREL:32

for R being Relation holds
( dom (CL R) c= dom R & rng (CL R) c= rng R & rng (CL R) c= dom R & dom (CL R) c= rng R )

proof end;

theorem :: SYSREL:33

for R being Relation holds
( id (dom (CL R)) c= id (dom R) & id (rng (CL R)) c= id (dom R) ) by Th15, Th32;

theorem Th34: :: SYSREL:34

for R being Relation holds
( id (dom (CL R)) c= R & id (rng (CL R)) c= R )

proof end;

theorem Th35: :: SYSREL:35

for X being set
for R being Relation holds
( ( id X c= R & (id X) * (R \ (id X)) = {} implies R | X = id X ) & ( id X c= R & (R \ (id X)) * (id X) = {} implies X |` R = id X ) )

proof end;

theorem :: SYSREL:36

for R being Relation holds
( ( (id (dom (CL R))) * (R \ (id (dom (CL R)))) = {} implies ( R | (dom (CL R)) = id (dom (CL R)) & R | (rng (CL R)) = id (dom (CL R)) ) ) & ( (R \ (id (rng (CL R)))) * (id (rng (CL R))) = {} implies ( (dom (CL R)) |` R = id (dom (CL R)) & (rng (CL R)) |` R = id (rng (CL R)) ) ) )

proof end;

theorem :: SYSREL:37

for R being Relation holds
( ( R * (R \ (id (dom R))) = {} implies (id (dom (CL R))) * (R \ (id (dom (CL R)))) = {} ) & ( (R \ (id (dom R))) * R = {} implies (R \ (id (dom (CL R)))) * (id (dom (CL R))) = {} ) )

proof end;

theorem Th38: :: SYSREL:38

for R, S being Relation holds
( ( S * R = S & R * (R \ (id (dom R))) = {} implies S * (R \ (id (dom R))) = {} ) & ( R * S = S & (R \ (id (dom R))) * R = {} implies (R \ (id (dom R))) * S = {} ) )

proof end;

theorem Th39: :: SYSREL:39

for R, S being Relation holds
( ( S * R = S & R * (R \ (id (dom R))) = {} implies CL S c= CL R ) & ( R * S = S & (R \ (id (dom R))) * R = {} implies CL S c= CL R ) )

proof end;

theorem :: SYSREL:40

for R, S being Relation holds
( ( S * R = S & R * (R \ (id (dom R))) = {} & R * S = R & S * (S \ (id (dom S))) = {} implies CL S = CL R ) & ( R * S = S & (R \ (id (dom R))) * R = {} & S * R = R & (S \ (id (dom S))) * S = {} implies CL S = CL R ) ) by Th39;