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Theorem List for Metamath Proof Explorer - 24501-24600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremordsuccl2 24501 If a successor of  A belongs to an ordinal, so does  A. (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( B  e.  On  /\ 
 suc  A  e.  B )  ->  A  e.  B )
 
Theoremordsuccl3 24502 If a successor of  A belongs to an ordinal,  A is a part of the ordinal. (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( B  e.  On  /\ 
 suc  A  e.  B )  ->  A  C_  B )
 
Theoremdomtri3 24503 Trichotomy law for dominance and strict dominance. This theorem is equivalent to the Axiom of Choice. (Contributed by FL, 16-Apr-2011.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( -.  A  ~<_  B  <->  B  ~<  A ) )
 
Theoremisfinite1b 24504 Omega strictly dominates a finite set. (Contributed by FL, 16-Apr-2011.)
 |-  ( A  e.  Fin  ->  A  ~<  om )
 
Theoremcptwff 24505 The cross product of two finite sets is finite. (Contributed by FL, 16-Apr-2011.)
 |-  (
 ( A  ~<  om  /\  B  ~<  om )  ->  ( A  X.  B )  ~<  om )
 
Theoreminttrp 24506 The intersection of a non-empty element of a transitive class is a part of the class. (Contributed by FL, 15-Apr-2011.)
 |-  (
 ( Tr  A  /\  B  e.  A  /\  B  =/=  (/) )  ->  |^| B  C_  A )
 
Theoremtrunitr 24507 The union of two transitive classes is transitive. JFM CLASSES1. th. 55 (Contributed by FL, 16-Apr-2011.)
 |-  (
 ( Tr  A  /\  Tr  B )  ->  Tr  ( A  u.  B ) )
 
Theoremuncum2 24508* Union of a cumulative hierarchy of sets. (Contributed by FL, 23-Apr-2011.)
 |-  ( A  e.  On  ->  U_ x  e.  A  ( R1 `  x ) 
 C_  ( R1 `  A ) )
 
Theoremcelsor 24509* If all the elements of a set  A are ordinal numbers and are parts of the set then  A is an ordinal number. (Contributed by FL, 20-Apr-2011.)
 |-  (
 ( A  e.  B  /\  A. x  e.  A  ( x  e.  On  /\  x  C_  A )
 )  ->  A  e.  On )
 
Theoremreflincror 24510 If a relation  R is reflexive, it is included in  ( R  o.  R
). (Contributed by FL, 8-May-2011.)
 |-  (
 ( Rel  R  /\  (  _I  |`  U. U. R )  C_  R )  ->  R  C_  ( R  o.  R ) )
 
Theoremfldrels 24511 The field of a relation is a set. (Contributed by FL, 23-May-2011.)
 |-  X  =  U. U. R   =>    |-  ( R  e.  S  ->  X  e.  _V )
 
Theoremfvsnn 24512 Value when  C doesn't belong to the domain. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  ( C  =/=  A  ->  ( { <. A ,  B >. } `  C )  =  (/) )
 
Theoremfvsn2a 24513 Value of a function with a domain of two different values. (Contributed by FL, 26-Jun-2011.)
 |-  A  e.  E   &    |-  B  e.  F   &    |-  C  e.  G   &    |-  D  e.  H   =>    |-  ( A  =/=  B  ->  ( { <. A ,  C >. ,  <. B ,  D >. } `  A )  =  C )
 
Theoremfvsn2b 24514 Value of a function with a domain of two different values. (Contributed by FL, 26-Jun-2011.)
 |-  A  e.  E   &    |-  B  e.  F   &    |-  C  e.  G   &    |-  D  e.  H   =>    |-  ( A  =/=  B  ->  ( { <. A ,  C >. ,  <. B ,  D >. } `  B )  =  D )
 
Theoremcnveq3 24515 A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.)
 |-  ( Rel  A  ->  ( A  =  (/)  <->  `' A  =  (/) ) )
 
Theoremrelrefcnv 24516 A relation is reflexive iff its converse is reflexive. (Contributed by FL, 19-Sep-2011.)
 |-  ( Rel  R  ->  ( (  _I  |`  U. U. R )  C_  R  <->  (  _I  |`  U. U. `' R )  C_  `' R ) )
 
Theoremeqfnung2 24517* If a family of sets  A indexed by  I covers the common domain  B of two functions  F and  G, the restrictions of  F and  G to  ( A  i^i  B ) are equal iff  F  =  G. Compare eqfnun 25786. (Contributed by FL, 5-Nov-2011.)
 |-  (
 ( B  C_  U_ i  e.  I  A  /\  F  Fn  B  /\  G  Fn  B )  ->  ( A. i  e.  I  ( F  |`  A )  =  ( G  |`  A )  <->  F  =  G )
 )
 
Theoreminjrec2 24518* A function is an injection iff a retraction exists. Bourbaki E.II.18 prop. 8. (Contributed by FL, 11-Nov-2011.)
 |-  (
 ( F : A --> B  /\  A  e.  C )  ->  ( F : A -1-1-> B  <->  E. r ( Fun  r  /\  ( r  o.  F )  =  (  _I  |`  A ) ) ) )
 
Theoremsurjsec2 24519* A function is an surjection iff a section exists. Bourbaki E.II.18 prop. 8. (Contributed by FL, 18-Nov-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
 |-  (
 ( F : A --> B  /\  A  e.  C  /\  B  e.  D ) 
 ->  ( F : A -onto-> B 
 <-> 
 E. s ( s : B --> A  /\  ( F  o.  s
 )  =  (  _I  |`  B ) ) ) )
 
Theoremab2rexexg2 24520* Existence of a class abstraction of existentially restricted sets. (Contributed by FL, 15-Oct-2012.)
 |-  (
 ( A  e.  D  /\  A. x  e.  A  B  e.  E )  ->  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }  e.  _V )
 
Theoremab2rexexg 24521* Existence of a class abstraction of existentially restricted sets. (Contributed by FL, 19-Apr-2012.)
 |-  (
 ( A  e.  D  /\  B  e.  E ) 
 ->  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }  e.  _V )
 
Theoremov2gc 24522* Value of a composition. ovmpt2g 5943 adapted to this special case of a composite. (Contributed by FL, 14-Jul-2012.)
 |-  O  =  ( x  e.  C ,  y  e.  D  |->  ( x  o.  y
 ) )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A O B )  =  ( A  o.  B ) )
 
Theoremov4gc 24523* Value of a composition. ovmpt4g 5931 adapted to the special case of a composite. (Contributed by FL, 14-Jul-2012.)
 |-  O  =  ( x  e.  C ,  y  e.  D  |->  ( x  o.  y
 ) )   =>    |-  ( ( x  e.  C  /\  y  e.  D )  ->  ( x O y )  =  ( x  o.  y
 ) )
 
Theoremdomintrefc 24524* The domain of the intersection of a family of reflexive classes is the intersection of the domains. (Contributed by FL, 15-Oct-2012.)
 |-  ( A. i  e.  A  A. x  e.  dom  R  x R x  ->  dom  |^|_  i  e.  A  R  =  |^|_ i  e.  A  dom  R )
 
Theoremrnintintrn 24525* The range of an intersection is a part of the intersection of the ranges. (The case  A  =  (/) works as well, the intersection gives  _V). (Contributed by FL, 15-Oct-2012.)
 |-  ran  |^|_ 
 x  e.  A  B  C_  |^|_ x  e.  A  ran  B
 
Theoremprjpacp1 24526 Projection of a part of a cross product. (Contributed by FL, 15-Oct-2012.)
 |-  (
 ( B  =/=  (/)  /\  C  C_  ( A  X.  B ) )  ->  ( 1st " C )  C_  A )
 
Theoremprjpacp2 24527 Projection of a part of a cross product. (Contributed by FL, 15-Oct-2012.)
 |-  (
 ( A  =/=  (/)  /\  C  C_  ( A  X.  B ) )  ->  ( 2nd " C )  C_  B )
 
Theoremrelinccppr 24528 A relation is included in the cross product of its projections. (Contributed by FL, 15-Oct-2012.)
 |-  ( Rel  A  ->  A  C_  (
 ( 1st " A )  X.  ( 2nd " A ) ) )
 
Theoremdffn5a 24529* Representation of a function in terms of its values. (Contributed by FL, 14-Sep-2013.) (Proof shortened by Mario Carneiro, 3-May-2015.)
 |-  F/_ x F   =>    |-  ( F  Fn  A  <->  F  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
 
Theoremffvelrnb 24530 A function's value belongs to its codomain. (Contributed by FL, 14-Sep-2013.)
 |-  (
 ( A  e.  D  /\  B  e.  E ) 
 ->  ( ( F  e.  ( B  ^m  A ) 
 /\  C  e.  A )  ->  ( F `  C )  e.  B ) )
 
Theoremab2rexex2g 24531* Existence of an existentially restricted class abstraction.  ph is normally has free-variable parameters  x,  y, and  z. Compare abrexex2g 5729. (Contributed by FL, 6-Nov-2013.)
 |-  (
 ( A  e.  _V  /\  B  e.  _V  /\  A. x  e.  A  A. y  e.  B  { z  |  ph }  e.  _V )  ->  { z  | 
 E. x  e.  A  E. y  e.  B  ph
 }  e.  _V )
 
Theoremfprg 24532 A function with a domain of two elements. (Contributed by FL, 2-Feb-2014.)
 |-  (
 ( ( A  e.  E  /\  B  e.  F )  /\  ( C  e.  G  /\  D  e.  H )  /\  A  =/=  B )  ->  { <. A ,  C >. ,  <. B ,  D >. } : { A ,  B } --> { C ,  D }
 )
 
Theoremiccss3 24533 Condition for a closed interval to be a subset of another closed interval. See iccss (Contributed by FL, 29-May-2014.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <_  C  /\  D  <_  B )
 )  ->  ( C [,] D )  C_  ( A [,] B ) )
 
Theoremiccleub2 24534 An element of a closed interval is more than or equal to its lower bound. (Contributed by FL, 29-May-2014.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A [,] B ) )  ->  A  <_  C )
 
Theoremiccleub3 24535 An element of a closed interval is less than or equal to its upper bound. (Contributed by FL, 29-May-2014.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A [,] B ) )  ->  C  e.  RR* )
 
Theoremxrre3 24536 A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B  <_  A  /\  A  <  +oo )
 )  ->  A  e.  RR )
 
Theoreminabs2 24537 Absorption law for intersection. (Contributed by FL, 30-May-2014.)
 |-  ( B  i^i  ( A  u.  B ) )  =  B
 
Theoreminttpemp 24538 Two ways of saying that two triples have no common element. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
 |-  (
 ( A  e.  G  /\  B  e.  H  /\  C  e.  I )  ->  ( ( { A ,  B ,  C }  i^i  { D ,  E ,  F } )  =  (/) 
 <->  ( ( A  =/=  D 
 /\  A  =/=  E  /\  A  =/=  F ) 
 /\  ( B  =/=  D 
 /\  B  =/=  E  /\  B  =/=  F ) 
 /\  ( C  =/=  D 
 /\  C  =/=  E  /\  C  =/=  F ) ) ) )
 
Theoremmapex2 24539* Two ways to express a subset of mappings. (Contributed by FL, 17-Nov-2014.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  { f  |  ( f : A --> B  /\  ph ) }  =  {
 f  e.  ( B 
 ^m  A )  | 
 ph } )
 
Theoremsssu 24540 Equality of a class difference and it subtrahend. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
 |-  (
 ( B  \  A )  =  A  <->  ( A  =  (/)  /\  B  =  (/) ) )
 
18.12.6  The "maps to" notation
 
Theoremcmpfun 24541 Functionality of a class given by a "maps to" notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |- 
 Fun  F
 
Theoremcmpdom 24542* Domain of a class given by the "maps to" notation. (Contributed by FL, 17-Feb-2008.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( A. x  e.  A  B  e.  _V  <->  dom  F  =  A )
 
Theoremcmpdom2 24543* Domain of a class given by the "maps to" notation. (Contributed by FL, 21-Jun-2010.)
 |-  F  =  ( x  e.  A  |->  ( B G C ) )   =>    |- 
 dom  F  =  A
 
Theoremfopab2g 24544* Functionality of an ordered-pair class abstraction given by the "maps to" notation. (Contributed by FL, 17-May-2010.) (Proof shortened by Mario Carneiro, 31-May-2014.)
 |-  ( F  =  ( x  e.  A  |->  C )  ->  ( A. x  e.  A  C  e.  B  <->  F : A --> B ) )
 
Theoremcrimmt1 24545* Composition of a restricted identity and a mapping (using the maps to notation). See fcoi1 5380. (Contributed by FL, 25-Apr-2012.)
 |-  ( F : A --> B  ->  ( F  o.  ( x  e.  A  |->  x ) )  =  F )
 
Theoremcrimmt2 24546* Composition of a restricted identity and a mapping (using the maps to notation). See fcoi2 5381. (Contributed by FL, 25-Apr-2012.)
 |-  ( F : A --> B  ->  ( ( x  e.  B  |->  x )  o.  F )  =  F )
 
Theoremmapmapmap 24547* Function returning a composite. (Contributed by FL, 19-Nov-2011.)
 |-  F1  =  ( f  e.  ( B  ^m  A )  |->  ( ( E  o.  f
 )  o.  G ) )   =>    |-  ( ( E : B
 --> B1  /\  G : A1 --> A  /\  ( ( A  e.  _V  /\  B  e.  _V )  /\  ( A1  e.  _V  /\  B1  e.  _V ) ) )  ->  F1 : ( B  ^m  A ) --> ( B1  ^m  A1 ) )
 
Theoreminjsurinj 24548* If  E is an injection and  G a surjection  ( f  |->  ( ( E  o.  f )  o.  G
) ) is an injection. Bourbaki E.II.31 prop. 2. (Contributed by FL, 20-Nov-2011.)
 |-  F1  =  ( f  e.  ( B  ^m  A )  |->  ( ( E  o.  f
 )  o.  G ) )   =>    |-  ( ( E : B -1-1-> B1  /\  G : A1
 -onto-> A  /\  ( ( A  e.  _V  /\  B  e.  _V )  /\  ( A1  e.  _V  /\  B1  e.  _V ) ) )  ->  F1 : ( B  ^m  A )
 -1-1-> ( B1  ^m  A1 )
 )
 
18.12.7  Cartesian Products

In what follows I will call nuple an element of a cartesian product.

If  X is a cartesian product,  N a nuple of  X,  I an indice,  ( ( X  pr  I ) `  N ) is the  I th coordinate of the nuple  N.

Suppose the set of indices is 
{ 1 ,  2 } and  X is the cartesian product  { { <. 1 ,  a >. , 
<. 2 ,  u >. } ,  { <. 1 ,  a >. , 
<. 2 ,  v
>. } } then  ( ( X  pr  1 ) `
 { <. 1 ,  a >. ,  <. 2 ,  u >. } )  =  a and  ( ( X  pr  2 ) `
 { <. 1 ,  a >. ,  <. 2 ,  u >. } )  =  u.

 
Syntaxcpro 24549 Extend class notation to include the projection mapping.
 class  pr
 
Syntaxcproj 24550 Extend class notation to include the projection mapping.
 class  prj
 
Definitiondf-pro 24551* Definition of a projection (also called a coordinate mapping). Meaninful if  x is a cartesian product and  y is an index. (Contributed by FL, 19-Jun-2011.)
 |-  pr  =  ( x  e.  _V ,  y  e.  _V  |->  ( f  e.  x  |->  ( f `  y
 ) ) )
 
Definitiondf-prj 24552* Definition of a projection. Meaninful if  x is a cartesian product and  y is a set of indices. Suppose  X  =  { { <. 1 ,  a
>. ,  <. 2 ,  u >. } ,  { <. 1 ,  a
>. ,  <. 2 ,  v >. } } then  ( X  prj  1 )  =  {
a } and  ( X  prj  2 )  =  {
u ,  v }. (Contributed by FL, 19-Jun-2011.)
 |-  prj  =  ( x  e.  _V ,  y  e.  _V  |->  ( f  e.  x  |->  ( f  |`  y ) ) )
 
Theoremelixp2b 24553* The base class of the elements of a nuple. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 12-Aug-2016.)
 |-  ( F  e.  X_ x  e.  A  B  ->  A. x  e.  A  ( F `  x )  e.  B )
 
Theorembclelnu 24554* The base class of an element of a nuple. (Contributed by FL, 19-Jun-2011.)
 |-  ( x  =  I  ->  B  =  C )   =>    |-  ( ( F  e.  X_ x  e.  A  B  /\  I  e.  A )  ->  ( F `  I )  e.  C )
 
Theoremispr1 24555* Definition of the coordinate mapping (or projection ) of index  I.  P is a cartesian product. (Contributed by FL, 19-Jun-2011.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  (
 ( P  e.  V  /\  I  e.  W )  ->  ( P  pr  I )  =  (
 f  e.  P  |->  ( f `  I ) ) )
 
Theoremprmapcp2 24556* A projection is a mapping from a cartesian product to an element of the family implied in the product. Bourbaki E.II.34 cor. 1. (Contributed by FL, 19-Jun-2011.)
 |-  P  =  X_ x  e.  A  B   &    |-  ( x  =  I 
 ->  B  =  C )   =>    |-  ( ( P  e.  V  /\  I  e.  A )  ->  ( P  pr  I ) : P --> C )
 
Theoremvalpr 24557 The  I th coordinate of the nuple  F. (Contributed by FL, 19-Jun-2011.)
 |-  (
 ( P  e.  V  /\  I  e.  W  /\  F  e.  P ) 
 ->  ( ( P  pr  I ) `  F )  =  ( F `  I ) )
 
Theoremnpincppr 24558* A set of nuples is included in the cartesian product of the projections of the nuples. Bourbaki E.II.32. (Contributed by FL, 20-Jun-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.)
 |-  P  =  X_ x  e.  A  B   =>    |-  ( ( F  C_  P  /\  P  e.  Q )  ->  F  C_  X_ x  e.  A  ( ( P  pr  x ) " F ) )
 
Theoremrepfuntw 24559 Representation as a set of pairs of a function whose domain has two distinct elements. (Contributed by FL, 26-Jun-2011.)
 |-  I  e.  A   &    |-  J  e.  B   =>    |-  ( I  =/=  J  ->  ( F  Fn  { I ,  J }  <->  F  =  { <. I ,  ( F `
  I ) >. , 
 <. J ,  ( F `
  J ) >. } ) )
 
Theoremrepcpwti 24560* A representation of a cartesian product with two indices. (Contributed by FL, 26-Jun-2011.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
 |-  A  =  { I ,  J }   &    |-  B  =  if ( x  =  I ,  M ,  N )   &    |-  I  e.  C   &    |-  J  e.  D   =>    |-  ( I  =/=  J  ->  X_ x  e.  A  B  =  {
 f  |  E. a  e.  M  E. b  e.  N  f  =  { <. I ,  a >. , 
 <. J ,  b >. } } )
 
Theoremcbcpcp 24561* The canonical bijection between a cross product and a cartesian product (whose set of indices is composed of two different elements). Bourbaki E.II.33 . (Contributed by FL, 26-Jun-2011.)
 |-  A  =  { I ,  J }   &    |-  B  =  if ( x  =  I ,  M ,  N )   &    |-  F  =  ( a  e.  M ,  b  e.  N  |->  {
 <. I ,  a >. , 
 <. J ,  b >. } )   &    |-  I  e.  C   &    |-  J  e.  D   =>    |-  ( I  =/=  J  ->  F : ( M  X.  N ) -1-1-onto-> X_ x  e.  A  B )
 
Theoremisprj1 24562* Definition of a projection.  I is a set of indices.  P is a cartesian product. (Contributed by FL, 19-Jun-2011.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  (
 ( P  e.  V  /\  I  e.  W )  ->  ( P  prj  I )  =  ( f  e.  P  |->  ( f  |`  I ) ) )
 
Theoremisprj2 24563* Definition of a projection.  I is a set of indices.  P is a cartesian product. (Contributed by FL, 19-Jun-2011.) (Revised by Mario Carneiro, 26-Jan-2015.)
 |-  P  =  X_ x  e.  A  B   =>    |-  ( ( I  e.  V  /\  A. x  e.  A  B  e.  D )  ->  ( P  prj  I )  =  ( f  e.  P  |->  ( f  |`  I ) ) )
 
Theoremprjmapcp 24564* A projection is a mapping from a cartesian product to one of its restriction. (Contributed by FL, 19-Jun-2011.) (Revised by Mario Carneiro, 26-Jan-2015.)
 |-  (
 ( I  C_  A  /\  A  e.  C  /\  A. x  e.  A  B  e.  D )  ->  ( X_ x  e.  A  B  prj  I ) : X_ x  e.  A  B --> X_ x  e.  I  B )
 
Theoremcbicp 24565* Canonical bijection between a cartesian product indexed by a singleton and the base class of the elements of the 1-uple. Bourbaki E II.32 (Contributed by FL, 6-Jun-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.)
 |-  ( x  =  A  ->  B  =  C )   =>    |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( X_ x  e.  { A } B  pr  A ) : X_ x  e.  { A } B -1-1-onto-> C )
 
Theoremprl 24566* Existence of a "prolongement" of a cartesian product. Bourbaki E.II.34 prop. 6. (Contributed by FL, 7-Nov-2011.)
 |-  A  e.  D   =>    |-  ( ( A. x  e.  A  C  =/=  (/)  /\  B  C_  A  /\  G  e.  X_ x  e.  B  C )  ->  E. f  e.  X_  x  e.  A  C G  C_  f )
 
Theoremprl1 24567* Existence of a "prolongement" of a cartesian product. Bourbaki E.II.34 prop. 6. (Contributed by FL, 20-Nov-2011.)
 |-  A  e.  D   =>    |-  ( ( A. x  e.  A  C  =/=  (/)  /\  B  C_  A )  ->  A. g  e.  X_  x  e.  B  C E. f  e.  X_  x  e.  A  C g  C_  f )
 
Theoremprl2 24568* Existence of a "prolongement" of a cartesian product. Bourbaki E.II.34 prop. 6. (Contributed by FL, 20-Nov-2011.)
 |-  A  e.  D   =>    |-  ( ( A. x  e.  A  C  =/=  (/)  /\  B  C_  A )  ->  A. g  e.  X_  x  e.  B  C E. f  e.  X_  x  e.  A  C g  =  ( f  |`  B ) )
 
Theoremprjmapcp2 24569* A projection is a mapping from a cartesian product onto one of its restriction. Bourbaki E.II.33 prop. 5. (Contributed by FL, 20-Nov-2011.) (Revised by Mario Carneiro, 31-May-2014.)
 |-  (
 ( I  C_  A  /\  A  e.  C  /\  ( A. x  e.  A  B  e.  D  /\  A. x  e.  A  B  =/= 
 (/) ) )  ->  ( X_ x  e.  A  B  prj  I ) :
 X_ x  e.  A  B -onto-> X_ x  e.  I  B )
 
Theoremdstr 24570* Distribution of union over intersection. Bourbaki E.II.35 prop. 8. (Contributed by FL, 18-Jun-2011.)
 |-  (
 y  =  ( f `
  x )  ->  C  =  D )   &    |-  X  =  X_ x  e.  A  B   &    |-  A  e.  E   =>    |-  U_ x  e.  A  |^|_
 y  e.  B  C  =  |^|_ f  e.  X  U_ x  e.  A  D
 
18.12.8  Operations on subsets and functions
 
Syntaxccst 24571 Extend class notation with an operator that derives an operation on subsets of a set from an operation on the elements of this set.
 class  cset
 
Definitiondf-cst 24572* Define an operation on the subsets derived from an operation  g on the elements. Meaningful if 
g is a binary internal operation. (Contributed by FL, 18-Apr-2010.)
 |-  cset  =  ( g  e.  _V  |->  ( x  e.  ~P dom  dom  g ,  y  e.  ~P dom  dom  g  |-> 
 ran  (  u  e.  x ,  v  e.  y  |->  ( u g v ) ) ) )
 
Theoremiscst1 24573* An operation on the subsets derived from an operation on the elements. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  X  =  dom  dom  G   &    |-  H  =  (
 cset `  G )   =>    |-  ( G  e.  A  ->  H  =  ( x  e.  ~P X ,  y  e.  ~P X  |->  ran  (  u  e.  x ,  v  e.  y  |->  ( u G v ) ) ) )
 
Theoremiscst2 24574* The value of the couple  <. A ,  B >. through the derived operation. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  X  =  dom  dom  G   &    |-  H  =  (
 cset `  G )   =>    |-  ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  ->  ( A H B )  =  ran  (  u  e.  A ,  v  e.  B  |->  ( u G v ) ) )
 
Theoremiscst3 24575* Property equivalent to the fact of belonging to the value of a pair through the derived operation. (Contributed by FL, 18-Apr-2010.)
 |-  X  =  dom  dom  G   &    |-  H  =  (
 cset `  G )   =>    |-  ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  ->  ( C  e.  ( A H B )  <->  E. u  e.  A  E. v  e.  B  C  =  ( u G v ) ) )
 
Theoremiscst4 24576* The value of the couple  <. A ,  B >. through the derived operation  H (expressed with a union). (Contributed by FL, 31-Dec-2010.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  X  =  dom  dom  G   &    |-  H  =  (
 cset `  G )   =>    |-  ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  ->  ( A H B )  =  U_ x  e.  B  ( A H { x } ) )
 
18.12.9  Arithmetic
 
Theorem3timesi 24577 Three times a number. (Contributed by FL, 17-Oct-2010.)
 |-  A  e.  CC   =>    |-  ( 3  x.  A )  =  ( A  +  ( A  +  A ) )
 
Theorem2eq3m1 24578  2 equals  3 minus  1. (Contributed by FL, 17-Oct-2010.)
 |-  2  =  ( 3  -  1
 )
 
TheoremnZdef 24579* Two ways to define  n ZZ. In the first way I multiply the set  { N } by the set  ZZ ( I think this is this sort of multiplication that is at the origin of the denotation  n ZZ). In the second way I multiply the integer  N by an element of  ZZ. (Contributed by FL, 18-Apr-2010.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
 |-  ( N  e.  ZZ  ->  ( { N }  ( cset `  (  x.  |`  ( ZZ 
 X.  ZZ ) ) ) ZZ )  =  { x  |  E. y  e.  ZZ  x  =  ( N  x.  y ) } )
 
18.12.10  Lattice (algebraic definition)
 
Syntaxclatalg 24580 Extend class notation to include the class of lattices.
 class  LatAlg
 
Definitiondf-latalg 24581* Algebraic definition of a lattice.  j is called the join of the lattice (and is denoted by 
\/ in textbooks) ,  m is called the meet (and is denoted by 
/\ in textbooks). (Contributed by FL, 11-Dec-2009.)
 |-  LatAlg  =  { <. j ,  m >.  | 
 E. t ( j : ( t  X.  t ) --> t  /\  m : ( t  X.  t ) --> t  /\  A. x  e.  t  A. y  e.  t  (
 ( x j y )  =  ( y j x )  /\  ( x m y )  =  ( y m x )  /\  (
 ( x m ( x j y ) )  =  x  /\  ( x j ( x m y ) )  =  x  /\  A. z  e.  t  (
 ( x m ( y m z ) )  =  ( ( x m y ) m z )  /\  ( x j ( y j z ) )  =  ( ( x j y ) j z ) ) ) ) ) }
 
Theoremislatalg 24582* The predicate "is a lattice". (Contributed by FL, 11-Dec-2009.)
 |-  X  =  dom  dom  J   =>    |-  ( ( J  e.  A  /\  M  e.  B )  ->  ( <. J ,  M >.  e.  LatAlg  <->  ( J :
 ( X  X.  X )
 --> X  /\  M :
 ( X  X.  X )
 --> X  /\  A. x  e.  X  A. y  e.  X  ( ( x J y )  =  ( y J x )  /\  ( x M y )  =  ( y M x )  /\  ( ( x M ( x J y ) )  =  x  /\  ( x J ( x M y ) )  =  x  /\  A. z  e.  X  ( ( x M ( y M z ) )  =  ( ( x M y ) M z )  /\  ( x J ( y J z ) )  =  ( ( x J y ) J z ) ) ) ) ) ) )
 
Theoremjop 24583 Join is a binary internal operation. (Contributed by FL, 12-Dec-2009.)
 |-  X  =  dom  dom  J   =>    |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  ->  J : ( X  X.  X ) --> X )
 
Theoremmop 24584 Meet is a binary internal operation. (Contributed by FL, 12-Dec-2009.)
 |-  X  =  dom  dom  J   =>    |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  ->  M : ( X  X.  X ) --> X )
 
Theoremcljo 24585 Closure of join. (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 Q )  e.  X )
 )
 
Theoremclme 24586 Closure of meet. (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 M Q )  e.  X )
 )
 
Theoremlabs1 24587* 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 M ( x J y ) )  =  x )
 
Theoremlabss1 24588 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 M ( P J Q ) )  =  P ) )
 
Theoremlabs2 24589* 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 24590 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 24591 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 24592 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 ) )
 
18.12.11  Currying and Partial Mappings
 
Syntaxccur1 24593 Extend class notation with the definition of currying.
 class  cur1
 
Syntaxccur2 24594 Extend class notation with the definition of currying.
 class  cur2
 
Definitiondf-cur1 24595* 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 24596* 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 24597* 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 24598* 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 24599* 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 24600 Lemma for domrancur1c 24601. (Contributed by FL, 17-May-2010.)
 |-  (
 ( F  Fn  ( A  X.  B )  /\  ( A  e.  C  /\  B  e.  D ) )  ->  ( F  o.  `' ( 2nd  |`  M ) )  e.  _V )
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