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Theorem List for Metamath Proof Explorer - 24601-24700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlabs2 24601* Absorption law.  ( x  \/  ( x  /\  y
) )  =  x. (Contributed by FL, 12-Dec-2009.)
 |-  X  =  dom  dom  J   =>    |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  ->  A. x  e.  X  A. y  e.  X  ( x J ( x M y ) )  =  x )
 
Theoremlabss2 24602 Absorption law.  ( P  \/  ( P  /\  Q ) )  =  P. (Contributed by FL, 12-Dec-2009.)
 |-  X  =  dom  dom  J   =>    |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  ->  (
 ( P  e.  X  /\  Q  e.  X ) 
 ->  ( P J ( P M Q ) )  =  P ) )
 
Theoremjidd 24603 Join is idempotent.  ( P  \/  P )  =  P. (Contributed by FL, 12-Dec-2009.)
 |-  X  =  dom  dom  J   =>    |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  ->  ( P  e.  X  ->  ( P J P )  =  P ) )
 
Theoremmidd 24604 Meet is idempotent.  ( P  /\  P )  =  P. (Contributed by FL, 12-Dec-2009.)
 |-  X  =  dom  dom  J   =>    |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  ->  ( P  e.  X  ->  ( P M P )  =  P ) )
 
18.13.11  Currying and Partial Mappings
 
Syntaxccur1 24605 Extend class notation with the definition of currying.
 class  cur1
 
Syntaxccur2 24606 Extend class notation with the definition of currying.
 class  cur2
 
Definitiondf-cur1 24607* Definition of currying (1st sort). Currying the operation  f consists in creating a mapping that returns for every value  x of  dom  dom  f the partial application of  f to  x. (Contributed by FL, 24-Jan-2010.)
 |-  cur1  =  { <. f ,  g >.  |  ( ( Fun  f  /\  Rel  dom  f )  /\  g  =  ( x  e.  dom  dom  f  |->  ( f  o.  `' ( 2nd  |`  ( { x }  X.  _V )
 ) ) ) ) }
 
Definitiondf-cur2 24608* Definition of currying (2nd sort). Currying the operation  f consists in creating a mapping that returns for every value  x of  ran  dom  f the partial application of  f to  x. (Contributed by FL, 24-Jan-2010.)
 |-  cur2  =  { <. f ,  g >.  |  ( Fun  f  /\  Rel  dom  f  /\  g  =  ( x  e.  ran  dom  f  |->  ( f  o.  `' ( 1st  |`  ( _V  X.  { x } ) ) ) ) ) }
 
Theoremcur1val 24609* The value of a curried operation. (Contributed by FL, 24-Jan-2010.)
 |-  (
 ( F  e.  A  /\  Fun  F  /\  Rel  dom 
 F )  ->  ( cur1 `  F )  =  ( x  e.  dom  dom 
 F  |->  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V )
 ) ) ) )
 
Theoremcur1vald 24610* The value of a curried operation. (Contributed by FL, 17-May-2010.)
 |-  (
 ( ( F  Fn  ( A  X.  B ) 
 /\  B  =/=  (/) )  /\  ( A  e.  C  /\  B  e.  D ) )  ->  ( cur1 `  F )  =  ( x  e.  A  |->  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) ) ) )
 
Theoremdomrancur1b 24611* The currying of a mapping  F whose domain is  ( A  X.  B
) is a mapping whose domain is  A and the range, the class of all the functions from  B to  ran  F. (Contributed by FL, 28-Apr-2010.)
 |-  A  e.  C   &    |-  B  e.  D   &    |-  B  =/= 
 (/)   &    |-  F  Fn  ( A  X.  B )   =>    |-  ( cur1 `  F ) : A --> { f  |  f : B --> ran  F }
 
Theoremdomrancur1clem 24612 Lemma for domrancur1c 24613. (Contributed by FL, 17-May-2010.)
 |-  (
 ( F  Fn  ( A  X.  B )  /\  ( A  e.  C  /\  B  e.  D ) )  ->  ( F  o.  `' ( 2nd  |`  M ) )  e.  _V )
 
Theoremdomrancur1c 24613* The currying of a mapping  F whose domain is  ( A  X.  B
) is a mapping whose domain is  A and the range, the class of all the functions from  B to  ran  F. (Contributed by FL, 17-May-2010.)
 |-  (
 ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B ) ) )  ->  ( cur1 `  F ) : A --> { f  |  f : B --> ran  F }
 )
 
Theoremvalcurfn 24614 The value of a curried function at 
O  e.  A is a mapping. (Contributed by FL, 17-May-2010.)
 |-  (
 ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B ) ) 
 /\  O  e.  A )  ->  ( ( cur1 `  F ) `  O ) : B --> ran  F )
 
Theoremvalcurfn1 24615 The value of a curried function at 
O  e.  A is a partial application. (Contributed by FL, 17-May-2010.)
 |-  G  =  ( F  o.  `' ( 2nd  |`  ( { O }  X.  _V ) ) )   =>    |-  ( ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B ) )  /\  O  e.  A )  ->  ( (
 cur1 `  F ) `  O )  =  G )
 
Theoremvalcurfn2 24616* The value of a curried function at 
O  e.  A is a partial application. (Contributed by FL, 17-May-2010.)
 |-  (
 ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B ) ) 
 /\  O  e.  A )  ->  ( ( cur1 `  F ) `  O )  =  ( x  e.  B  |->  ( O F x ) ) )
 
Theoremvalvalcurfn 24617 The value at  P  e.  B of the value of a curried function at  O  e.  A equals  ( O F P ). (Contributed by FL, 17-May-2010.)
 |-  (
 ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B ) ) 
 /\  ( O  e.  A  /\  P  e.  B ) )  ->  ( ( ( cur1 `  F ) `  O ) `  P )  =  ( O F P ) )
 
18.13.12  Order theory (Extensible Structure Builder)
 
Syntaxcorhom 24618 Extend class notation with the class of all decreasing functions.
 class  OrHom
 
Syntaxcoriso 24619 Extend class notation with the class of all the order isomorphisms.
 class  OrIso
 
Definitiondf-orhom 24620* Increasing functions also called "order homomorphisms", "isotone, monotone or order preserving mappings". To have the class of decreasing functions use  ( r  OrHom  `' s ). Experimental. Bourbaki E.III.7 (Contributed by FL, 17-Nov-2014.)
 |-  OrHom  =  ( r  e.  _V ,  s  e.  _V  |->  { f  e.  ( ( Base `  s
 )  ^m  ( Base `  r ) )  | 
 A. a  e.  ( Base `  r ) A. b  e.  ( Base `  r ) ( a ( le `  r
 ) b  ->  (
 f `  a )
 ( le `  s
 ) ( f `  b ) ) }
 )
 
Definitiondf-oriso 24621* Order isomorphisms. Experimental. Bourbaki E.III.5 (Contributed by FL, 17-Nov-2014.)
 |-  OrIso  =  ( r  e.  _V ,  s  e.  _V  |->  { f  |  ( f : (
 Base `  r ) -1-1-onto-> ( Base `  s )  /\  A. a  e.  ( Base `  r ) A. b  e.  ( Base `  r )
 ( a ( le `  r ) b  <->  ( f `  a ) ( le `  s ) ( f `
  b ) ) ) } )
 
Theoremisorhom 24622* The predicate is an order homomorphism. (Contributed by FL, 17-Nov-2014.)
 |-  X  =  ( Base `  A )   &    |-  Y  =  ( Base `  B )   &    |- &lea  =  ( le `  A )   &    |- &leb  =  ( le `  B )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  OrHom  B )  =  { f  e.  ( Y  ^m  X )  |  A. a  e.  X  A. b  e.  X  ( a&lea  b 
 ->  ( f `  a
 )&leb  ( f `  b ) ) }
 )
 
Theoremisoriso 24623* Order isomorphisms. (Contributed by FL, 17-Nov-2014.)
 |-  X  =  ( Base `  A )   &    |-  Y  =  ( Base `  B )   &    |- &lea  =  ( le `  A )   &    |- &leb  =  ( le `  B )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  OrIso  B )  =  { f  |  ( f : X -1-1-onto-> Y  /\  A. a  e.  X  A. b  e.  X  ( a&lea  b  <->  ( f `  a )&leb  ( f `  b ) ) ) } )
 
Theoremisoriso2 24624* Order isomorphisms. (Contributed by FL, 17-Nov-2014.)
 |-  X  =  ( Base `  A )   &    |-  Y  =  ( Base `  B )   &    |- &lea  =  ( le `  A )   &    |- &leb  =  ( le `  B )   =>    |-  ( ( A  e.  C  /\  B  e.  D  /\  F  e.  E ) 
 ->  ( F  e.  ( A  OrIso  B )  <->  ( F : X
 -1-1-onto-> Y  /\  A. a  e.  X  A. b  e.  X  ( a&lea  b  <-> 
 ( F `  a
 )&leb  ( F `  b ) ) ) ) )
 
Theoremoriso 24625 If  F is an order isomorphism so is  `' F. (Contributed by FL, 11-Nov-2014.)
 |-  X  =  ( Base `  A )   &    |-  Y  =  ( Base `  B )   &    |- &lea  =  ( le `  A )   &    |- &leb  =  ( le `  B )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( F  e.  ( A  OrIso  B ) 
 ->  `' F  e.  ( B  OrIso  A ) ) )
 
18.13.13  Order theory
 
Syntaxcpresetrel 24626 Extend class notation with the class of all the presets.
 class PresetRel
 
Syntaxcmxl 24627 Extend class notation with a function that returns the maximal elements of a preset.
 class  mxl
 
Syntaxcmnl 24628 Extend class notation with a function that returns the minimal elements of a preset.
 class  mnl
 
Syntaxcub 24629 Extend class notation with a function that returns the upper bounds of a part of a preset.
 class  ub
 
Syntaxclb 24630 Extend class notation with a function that returns the lower bounds of a part of a preset.
 class  lb
 
Syntaxcge 24631 Extend class notation with a function that returns the greatest element of a poset.
 class  ge
 
Syntaxcse 24632 Extend class notation with a function that returns the smallest element of a poset.
 class leR
 
Syntaxcantidir 24633 Extend class notation with the class of all the anti-directions.
 class  AntiDir
 
Definitiondf-prs 24634 Define the class of all presets. A preset is a transitive and reflexive relation. ("preset" is the short for preordered set.) (Contributed by FL, 1-May-2011.)
 |- PresetRel  =  {
 r  |  ( Rel  r  /\  ( r  o.  r )  C_  r  /\  (  _I  |`  U. U. r )  C_  r ) }
 
Theoremisprsr 24635 The predicate "is a preset". (Contributed by FL, 1-May-2011.)
 |-  ( R  e.  A  ->  ( R  e. PresetRel  <->  ( Rel  R  /\  ( R  o.  R )  C_  R  /\  (  _I  |`  U. U. R )  C_  R ) ) )
 
Theorempreorel 24636 A preset is a relation. (Contributed by FL, 18-May-2011.)
 |-  ( R  e. PresetRel  ->  Rel  R )
 
Theorempreodom2 24637 The domain of a preset equals its field. (Contributed by FL, 22-May-2011.)
 |-  ( R  e. PresetRel  ->  dom  R  =  U.
 U. R )
 
Theoremppldrels 24638 The field of a preset is a set. (Contributed by FL, 19-Sep-2011.)
 |-  X  =  dom  R   =>    |-  ( R  e. PresetRel  ->  X  e.  _V )
 
Theorempreoref12 24639 A preset is reflexive. (Contributed by FL, 18-May-2011.)
 |-  X  =  dom  R   =>    |-  ( R  e. PresetRel  ->  (  _I  |`  X )  C_  R )
 
Theorempreoref22 24640 A preset is reflexive. (Contributed by FL, 22-May-2011.)
 |-  X  =  dom  R   =>    |-  ( ( R  e. PresetRel  /\  A  e.  X )  ->  A R A )
 
Theorempreoran2 24641 The range of a preset equals its field. (Contributed by FL, 22-May-2011.)
 |-  X  =  dom  R   =>    |-  ( R  e. PresetRel  ->  ran  R  =  X )
 
Theorempre1befi2 24642 If  A  <_  B then 
A belongs to the field of the preset. (Contributed by FL, 23-May-2011.)
 |-  X  =  dom  R   =>    |-  ( ( R  e. PresetRel  /\  A R B )  ->  A  e.  X )
 
Theorempre2befi2 24643 If  A  <_  B then 
B belongs to the field of the preset. (Contributed by FL, 23-May-2011.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  X  =  dom  R   =>    |-  ( ( R  e. PresetRel  /\  A R B )  ->  B  e.  X )
 
Theoremdomcnvpre 24644 If  R is a preset, its domain and the domain of its converse are equal. (Contributed by FL, 19-Sep-2011.)
 |-  X  =  dom  R   =>    |-  ( R  e. PresetRel  ->  X  =  dom  `'  R )
 
Theorempreotr1 24645 A preset is transitive. (Contributed by FL, 22-May-2011.)
 |-  ( R  e. PresetRel  ->  ( R  o.  R )  C_  R )
 
Theorempreotr2 24646 A preset is transitive. (Contributed by FL, 23-May-2011.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  (
 ( R  e. PresetRel  /\  ( A R B  /\  B R C ) )  ->  A R C )
 
Theoremaltprs2 24647 The composite of a preset with itself. (Contributed by FL, 13-May-2011.)
 |-  ( R  e. PresetRel  ->  ( R  o.  R )  =  R )
 
Theoremint2pre 24648 The intersection of two presets is a preset. (Contributed by FL, 28-Dec-2011.)
 |-  (
 ( R  e. PresetRel  /\  S  e. PresetRel )  ->  ( R  i^i  S )  e. PresetRel )
 
Theoremsqpre 24649 A square product is a preset. (Contributed by FL, 28-Dec-2011.)
 |-  ( A  e.  V  ->  ( A  X.  A )  e. PresetRel )
 
Theoremindpre 24650 The relation induced by a preset on a part of its field is a preset. (Contributed by FL, 28-Dec-2011.)
 |-  (
 ( R  e. PresetRel  /\  A  e.  B )  ->  ( R  i^i  ( A  X.  A ) )  e. PresetRel )
 
Theoremposprsr 24651 A partial order is a preset. (Contributed by FL, 1-May-2011.)
 |-  PosetRel  C_ PresetRel
 
Theoremposispre 24652 A poset is a preset. (Contributed by FL, 19-Sep-2011.)
 |-  ( A  e.  PosetRel  ->  A  e. PresetRel )
 
Theoremempos 24653 The empty set is a poset. (Contributed by FL, 6-Oct-2008.)
 |-  (/)  e.  PosetRel
 
Theoremdupre1 24654 The converse of a preset is a preset. The case  ( `' R  e. PresetRel  ->  R  e. PresetRel ) is true only if 
R is a relation. See dupre2 24655. (Contributed by FL, 5-Jan-2009.)
 |-  ( R  e. PresetRel  ->  `' R  e. PresetRel )
 
Theoremdupre2 24655 The converse of a preset is a preset. (Contributed by FL, 19-Sep-2011.)
 |-  ( Rel  R  ->  ( R  e. PresetRel  <->  `' R  e. PresetRel ) )
 
Theoremnfwval 24656 An infimum is the supremum of the converse relation. (Contributed by FL, 6-Sep-2009.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  (
 ( R  e.  U  /\  A  e.  W ) 
 ->  ( R  inf w  A )  =  ( `' R  sup w  A ) )
 
Definitiondf-mxl 24657* Define the maximal elements of a set. I.e. the elements of the set that are not smaller than the other elements. Meaningful if  r is at least a preset. Read  ( mxl `  R
) as the maximal elements of the set  U. U. R preordered by  R. Bourbaki E III 8. Experimental. (Contributed by FL, 16-May-2011.)
 |-  mxl  =  ( r  e.  _V  |->  { a  e.  U. U. r  |  A. b  e. 
 U. U. r ( a r b  ->  a  =  b ) } )
 
Definitiondf-mnl 24658* Define the minimal elements of a set. I.e. the elements of the set that are not greater than the other elements. Meaningful is  r is at least a preset. Read  ( mnl `  R
) as the minimal elements of the set  U. U. R preordered by  R. Bourbaki E III 8. Experimental. (Contributed by FL, 19-Sep-2011.)
 |-  mnl  =  ( r  e.  _V  |->  { a  e.  U. U. r  |  A. b  e. 
 U. U. r ( b r a  ->  b  =  a ) } )
 
Definitiondf-ge 24659 Define the greatest element of a poset. I.e. the element of the poset that is larger than the other elements. Meaningful is  r is at least a poset (otherwise there could be more than one supremum due to cycles). Bourbaki E III 10. Experimental. (Contributed by FL, 19-Sep-2011.)
 |-  ge  =  ( r  e.  _V  |->  ( r  sup w  dom  r ) )
 
Definitiondf-ler 24660 Define the least element of a poset. I.e. the element of the poset that is smaller than the other elements. Meaningful is  r is at least a poset. Experimental. (Contributed by FL, 19-Sep-2011.)
 |- leR  =  ( r  e.  _V  |->  ( r  inf w  dom  r ) )
 
Theoremgepsup 24661 The greatest element of a poset is the supremum of the poset. (Contributed by FL, 19-Sep-2011.)
 |-  X  =  dom  R   =>    |-  ( R  e.  A  ->  ( ge `  R )  =  ( R  sup w  X ) )
 
Theoremseinf 24662 The least element of a poset is the infimum of the poset. (Contributed by FL, 19-Sep-2011.)
 |-  X  =  dom  R   =>    |-  ( R  e.  A  ->  (leR `  R )  =  ( R  inf w  X ) )
 
Theoremsege 24663 The least element of a poset is the greatest element of the converse poset. (Contributed by FL, 30-Dec-2011.)
 |-  ( R  e.  PosetRel  ->  (leR `  R )  =  ( ge `  `' R ) )
 
Definitiondf-ub 24664* Define the upper bounds of a set  x. Meaningful if  r is at least a preset, and  x a subset of the field of 
r. Bourbaki E.III.9 def. 5. Experimental. (Contributed by FL, 16-May-2011.)
 |-  ub  =  ( r  e.  _V ,  x  e.  _V  |->  { a  e.  U. U. r  |  A. b  e.  x  b r a } )
 
Definitiondf-lb 24665* Define the lower bounds of a set  x. Meaningful if  r is at least a preset, and  x a subset of the field of 
r. Experimental. (Contributed by FL, 16-May-2011.)
 |-  lb  =  ( r  e.  _V ,  x  e.  _V  |->  { a  e.  U. U. r  |  A. b  e.  x  a r b } )
 
Definitiondf-antidir 24666* An antidirected set (also called a set filtering on the left by Bourbaki) is a preset whose every pair of elements has a lower bound. (Contributed by FL, 17-Oct-2011.)
 |-  AntiDir  =  (PresetRel  i^i  { r  |  A. x  e.  U. U. r A. y  e.  U. U. r E. z  e.  U. U. r z  e.  (
 r lb { x ,  y } ) }
 )
 
Theoremubos 24667* The upper bounds of  A. (Contributed by FL, 16-May-2011.)
 |-  X  =  U. U. R   =>    |-  ( ( R  e.  S  /\  A  e.  B )  ->  ( R  ub  A )  =  { a  e.  X  |  A. b  e.  A  b R a } )
 
Theoremubos2 24668* The upper bounds of  A. (Contributed by FL, 18-Sep-2011.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  X  =  dom  R   =>    |-  ( ( R  e. PresetRel  /\  A  e.  B )  ->  ( R  ub  A )  =  { a  e.  X  |  A. b  e.  A  b R a } )
 
Theorempuub2 24669* The predicate " U is an upper bound of  A." (Contributed by FL, 16-May-2011.)
 |-  X  =  dom  R   =>    |-  ( ( R  e. PresetRel  /\  A  e.  B )  ->  ( U  e.  ( R  ub  A )  <->  ( U  e.  X  /\  A. b  e.  A  b R U ) ) )
 
Theorempuub 24670* The predicate " U is an upper bound of  A." (Contributed by FL, 16-May-2011.)
 |-  X  =  U. U. R   =>    |-  ( ( R  e.  S  /\  A  e.  B )  ->  ( U  e.  ( R  ub  A )  <->  ( U  e.  X  /\  A. b  e.  A  b R U ) ) )
 
Theoremprltub 24671 If  R is a preset,  U R V and  U is an upper bound of  A then  V is an upper bound of  A. Bourbaki E.III.9 nb 8. (Contributed by FL, 23-May-2011.) (Proof shortened by Mario Carneiro, 3-May-2015.)
 |-  (
 ( R  e. PresetRel  /\  A  e.  B  /\  ( U  e.  ( R  ub  A )  /\  U R V ) )  ->  V  e.  ( R  ub  A ) )
 
Theoremubpar 24672 If  U is an upper bound of  A and  B  C_  A then  U is an upper bound of  B. Bourbaki E.III.9 n 8. (Contributed by FL, 23-May-2011.)
 |-  (
 ( R  e. PresetRel  /\  A  e.  C  /\  B  C_  A )  ->  ( U  e.  ( R  ub  A )  ->  U  e.  ( R  ub  B ) ) )
 
Theoremsupdef 24673* If it exists, a supremum of  A is greater or equal to every element of  A and is the least upper bound of  A. Here the existence of the supremum is expressed by the idiom  ( R  sup w  A
)  e.  X. (Contributed by FL, 23-May-2011.)
 |-  X  =  dom  R   =>    |-  ( ( R  e.  PosetRel  /\  A  e.  W  /\  ( R  sup w  A )  e.  X )  ->  ( A. y  e.  A  y R ( R  sup w  A )  /\  A. y  e.  X  ( A. z  e.  A  z R y 
 ->  ( R  sup w  A ) R y ) ) )
 
Theoremsupdefa 24674 The greatest element of a poset is greater than the other elements of the poset. (Contributed by FL, 19-Sep-2011.)
 |-  X  =  dom  R   =>    |-  ( ( R  e.  PosetRel  /\  A  e.  X  /\  ( R  sup w  X )  e.  X )  ->  A R ( R 
 sup w  X )
 )
 
Theoremmxlelt 24675* The maximal elements of the preset 
R. (Contributed by FL, 16-May-2011.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. U. R   =>    |-  ( R  e.  S  ->  ( mxl `  R )  =  { a  e.  X  |  A. b  e.  X  ( a R b  ->  a  =  b ) } )
 
Theoremmxlelt2 24676* The maximal elements of the preset 
R. (Contributed by FL, 16-May-2011.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  dom  R   =>    |-  ( R  e. PresetRel  ->  ( mxl `  R )  =  { a  e.  X  |  A. b  e.  X  ( a R b 
 ->  a  =  b
 ) } )
 
Theoremmnlelt2 24677* The minimal elements of the preset 
R. (Contributed by FL, 19-Sep-2011.)
 |-  X  =  dom  R   =>    |-  ( R  e. PresetRel  ->  ( mnl `  R )  =  { a  e.  X  |  A. b  e.  X  ( b R a 
 ->  b  =  a
 ) } )
 
Theoremismxl2 24678* The predicate " A is a maximal element of the preset 
R " . (Contributed by FL, 22-May-2011.)
 |-  X  =  dom  R   =>    |-  ( R  e. PresetRel  ->  ( A  e.  ( mxl `  R )  <->  ( A  e.  X  /\  A. b  e.  X  ( A R b  ->  A  =  b ) ) ) )
 
Theoremismnl2 24679* The predicate " A is a minimal element of the preset 
R " . (Contributed by FL, 19-Sep-2011.)
 |-  X  =  dom  R   =>    |-  ( R  e. PresetRel  ->  ( A  e.  ( mnl `  R )  <->  ( A  e.  X  /\  A. b  e.  X  ( b R A  ->  b  =  A ) ) ) )
 
Theoremmnlmxl2 24680 The minimal elements of a preset are the maximal elements of the converse preset. (Contributed by FL, 19-Sep-2011.)
 |-  ( R  e. PresetRel  ->  ( mnl `  R )  =  ( mxl `  `' R ) )
 
Theoremmxlmnl2 24681 The maximal elements of a preset are the minimal elements of the converse preset. (Contributed by FL, 19-Sep-2011.)
 |-  ( R  e. PresetRel  ->  ( mxl `  R )  =  ( mnl `  `' R ) )
 
Theoremdefge3 24682* The greatest element of a poset is an element, when it exists, that is greater than the other elements of the poset. Use the idiom  ( ge `  R )  e.  X when you mean the greatest element of  X exists. (Contributed by FL, 30-Dec-2011.)
 |-  X  =  dom  R   =>    |-  ( ( R  e.  PosetRel  /\  ( ge `  R )  e.  X )  ->  A. x  e.  X  x R ( ge `  R ) )
 
Theoremdefse3 24683* The least element of a poset is an element, when it exists, that is less than the other elements of the poset. Use the idiom  (leR `  R
)  e.  X when you mean the least element of  X exists. (Contributed by FL, 30-Dec-2011.)
 |-  X  =  dom  R   =>    |-  ( ( R  e.  PosetRel  /\  (leR `  R )  e.  X )  ->  A. x  e.  X  (leR `  R ) R x )
 
Theoremsupaub 24684 If it exists, a supremum of  A is an upper bound for  A. (Contributed by FL, 19-Sep-2011.)
 |-  X  =  dom  R   =>    |-  ( ( R  e.  PosetRel  /\  A  e.  W  /\  ( R  sup w  A )  e.  X )  ->  ( R  sup w  A )  e.  ( R  ub  A ) )
 
Theoremsupwlub 24685* If it exists, a supremum of  A is the least upper bound for  A. (Contributed by FL, 19-Sep-2011.)
 |-  X  =  dom  R   =>    |-  ( ( R  e.  PosetRel  /\  A  e.  W  /\  ( R  sup w  A )  e.  X )  ->  A. x  e.  ( R  ub  A ) ( R  sup w  A ) R x )
 
Theoremgeme2 24686 The greatest element of  X is a maximal element. (Contributed by FL, 19-Sep-2011.)
 |-  X  =  dom  R   =>    |-  ( ( R  e.  PosetRel  /\  ( R  sup w  X )  e.  X )  ->  ( R  sup w  X )  e.  ( mxl `  R ) )
 
Theoreminposetlem 24687* Lemma for inposet 24689. Definition of inclusion of sets using a class. (Contributed by FL, 22-Sep-2008.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  =  { <. x ,  y >.  |  x  C_  y }   =>    |-  ( A C B  <->  A 
 C_  B )
 
Theoreminpc 24688* Inclusion is a proper class. (Contributed by FL, 22-Sep-2008.)
 |-  C  =  { <. x ,  y >.  |  x  C_  y }   =>    |- 
 -.  C  e.  _V
 
Theoreminposet 24689* Inclusion partially orders any set. (Contributed by FL, 22-Sep-2008.)
 |-  C  =  { <. x ,  y >.  |  x  C_  y }   =>    |-  ( A  e.  B  ->  ( C  i^i  ( A  X.  A ) )  e.  PosetRel )
 
Theoremdefinc 24690* Definition of the inclusion. (Contributed by FL, 6-Sep-2009.)
 |-  G  e.  X   &    |-  F  e.  Y   &    |-  C  =  { <. x ,  y >.  |  x  C_  y }   =>    |-  ( G ( C  i^i  ( A  X.  B ) ) F  <-> 
 ( G  e.  A  /\  F  e.  B  /\  G  C_  F ) )
 
Theoremdominc 24691* The domain of the inclusion relation is  _V. (Contributed by FL, 6-Sep-2009.)
 |-  C  =  { <. x ,  y >.  |  x  C_  y }   =>    |- 
 dom  C  =  _V
 
Theoremrninc 24692* The range of the inclusion relation is  _V. (Contributed by FL, 6-Sep-2009.)
 |-  C  =  { <. x ,  y >.  |  x  C_  y }   =>    |- 
 ran  C  =  _V
 
Theoremdomncnt 24693* Domain of the intersection of the inclusion with a square cross product. (Contributed by FL, 6-Sep-2009.)
 |-  C  =  { <. x ,  y >.  |  x  C_  y }   =>    |- 
 dom  (  C  i^i  ( A  X.  A ) )  =  A
 
Theoremranncnt 24694* Range of the intersection of the inclusion with a square cross product. (Contributed by FL, 6-Sep-2009.)
 |-  C  =  { <. x ,  y >.  |  x  C_  y }   =>    |- 
 ran  (  C  i^i  ( A  X.  A ) )  =  A
 
Theoremsupwval 24695 Value of an infimum under a weak ordering. (Contributed by FL, 19-Sep-2011.)
 |-  (
 ( Rel  R  /\  R  e.  U  /\  A  e.  W )  ->  ( R  sup w  A )  =  ( `' R  inf w  A ) )
 
Theoremnfwpr4c 24696 Infimum of an unordered pair of comparable elements. (Contributed by FL, 19-Sep-2011.)
 |-  (
 ( R  e.  PosetRel  /\  A R B )  ->  ( R  inf w  { A ,  B }
 )  =  A )
 
Theoremtolat 24697 A totally ordered set is a lattice. (Contributed by FL, 19-Sep-2011.)
 |-  TosetRel  C_  LatRel
 
Theoremdispos 24698 A restriction of the identity is a poset. (Contributed by FL, 2-Aug-2009.)
 |-  ( A  e.  V  ->  (  _I  |`  A )  e. 
 PosetRel )
 
Theoremderef 24699 An idiom to "dereflexivate" a relation. (Contributed by FL, 30-Dec-2010.)
 |-  ( A  e.  B  ->  -.  A ( C  \  _I  ) A )
 
Theoremdfps2 24700 Alternate definition of a poset. Bourbaki E.III.2 prop. 1. (Contributed by FL, 30-Dec-2010.)
 |-  PosetRel  =  {
 r  |  ( Rel  r  /\  ( r  o.  r )  =  r  /\  ( r  i^i  `' r )  =  (  _I  |`  U. U. r ) ) }
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