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Theorem List for Metamath Proof Explorer - 23801-23900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcfullfn 23801 Declare the syntax for the full function functor.
 class FullFun F
 
Syntaxcrestrict 23802 Declare the syntax for the restriction function.
 class Restrict
 
Definitiondf-txp 23803 Define the tail cross of two classes. Membership in this class is defined by txpss3v 23826 and brtxp 23828. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  ( A  (x)  B )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B ) )
 
Definitiondf-pprod 23804 Define the parallel product of two classes. Membership in this class is defined by pprodss4v 23832 and brpprod 23833. (Contributed by Scott Fenton, 11-Apr-2014.)
 |- pprod ( A ,  B )  =  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )
 
Definitiondf-sset 23805 Define the subset class. For the value, see brsset 23837. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  SSet  =  ( ( _V  X.  _V )  \  ran  (  _E  (x)  ( _V  \  _E  ) ) )
 
Definitiondf-trans 23806 Define the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  Trans  =  ( _V  \  ran  (
 (  _E  o.  _E  )  \  _E  ) )
 
Definitiondf-bigcup 23807 Define the Bigcup function, which, per fvbigcup 23850, carries a set to its union. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  Bigcup  =  ( ( _V  X.  _V )  \  ran  ( ( _V  (x)  _E  )(++) ( (  _E  o.  _E  )  (x)  _V ) ) )
 
Definitiondf-fix 23808 Define the class of all fixpoints of a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  Fix A  =  dom  (  A  i^i  _I  )
 
Definitiondf-limits 23809 Define the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  Limits  =  ( ( On  i^i  Fix Bigcup )  \  { (/) } )
 
Definitiondf-funs 23810 Define the class of all functions. See elfuns 23861 for membership. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  Funs  =  ( ~P ( _V 
 X.  _V )  \  Fix (  _E  o.  ( ( 1st  (x)  ( ( _V  \  _I  )  o. 
 2nd ) )  o.  `'  _E  ) ) )
 
Definitiondf-singleton 23811 Define the singleton function. See brsingle 23863 for its value. (Contributed by Scott Fenton, 4-Apr-2014.)
 |- Singleton  =  ( ( _V  X.  _V )  \  ran  ( ( _V  (x)  _E  )(++) (  _I  (x)  _V )
 ) )
 
Definitiondf-singles 23812 Define the class of all singletons. See elsingles 23864 for membership. (Contributed by Scott Fenton, 19-Feb-2013.)
 |-  Singletons  =  ran Singleton
 
Definitiondf-image 23813 Define the image functor. This function takes a set  A to a function  x  |->  ( A
" x ), providing that the latter exists. See imageval 23876 for the derivation. (Contributed by Scott Fenton, 27-Mar-2014.)
 |- Image A  =  ( ( _V  X.  _V )  \  ran  (
 ( _V  (x)  _E  )(++) ( (  _E  o.  `' A )  (x)  _V )
 ) )
 
Definitiondf-cart 23814 Define the cartesian product function. See brcart 23878 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
 |- Cart  =  ( ( ( _V  X.  _V )  X.  _V )  \ 
 ran  ( ( _V 
 (x)  _E  )(++) (pprod (  _E  ,  _E  )  (x)  _V ) ) )
 
Definitiondf-img 23815 Define the image function. See brimg 23883 for its value. (Contributed by Scott Fenton, 12-Apr-2014.)
 |- Img  =  (Image ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) )  o. Cart
 )
 
Definitiondf-domain 23816 Define the domain function. See brdomain 23879 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
 |- Domain  = Image ( 1st  |`  ( _V  X.  _V ) )
 
Definitiondf-range 23817 Define the range function. See brrange 23880 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
 |- Range  = Image ( 2nd  |`  ( _V  X.  _V ) )
 
Definitiondf-cup 23818 Define the little cup function. See brcup 23885 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
 |- Cup  =  ( ( ( _V  X.  _V )  X.  _V )  \ 
 ran  ( ( _V 
 (x)  _E  )(++) (
 ( ( `' 1st  o. 
 _E  )  u.  ( `' 2nd  o.  _E  )
 )  (x)  _V )
 ) )
 
Definitiondf-cap 23819 Define the little cap function. See brcap 23886 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
 |- Cap  =  ( ( ( _V  X.  _V )  X.  _V )  \ 
 ran  ( ( _V 
 (x)  _E  )(++) (
 ( ( `' 1st  o. 
 _E  )  i^i  ( `' 2nd  o.  _E  )
 )  (x)  _V )
 ) )
 
Definitiondf-restrict 23820 Define the restriction function. See brrestrict 23894 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
 |- Restrict  =  (Cap 
 o.  ( 1st  (x)  (Cart  o.  ( 2nd  (x)  (Range  o. 
 1st ) ) ) ) )
 
Definitiondf-succf 23821 Define the successor function. See brsuccf 23887 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
 |- Succ  =  (Cup 
 o.  (  _I  (x) Singleton ) )
 
Definitiondf-apply 23822 Define the application function. See brapply 23884 for its value. (Contributed by Scott Fenton, 12-Apr-2014.)
 |- Apply  =  ( ( Bigcup  o.  Bigcup )  o.  ( ( ( _V 
 X.  _V )  \  ran  ( ( _V  (x)  _E  )(++) ( (  _E  |` 
 Singletons )  (x)  _V )
 ) )  o.  (
 (Singleton  o. Img )  o. pprod (  _I  , Singleton ) ) ) )
 
Definitiondf-funpart 23823 Define the functional part of a class  F. This is the maximal part of  F that is a function. See funpartfun 23888 and funpartfv 23890 for the meaning of this statement. (Contributed by Scott Fenton, 16-Apr-2014.)
 |- Funpart F  =  ( F  |`  dom  (
 (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) )
 
Definitiondf-fullfun 23824 Define the full function over  F. This is a function with domain  _V that always agrees with  F for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
 |- FullFun F  =  (Funpart F  u.  ( ( _V  \  dom Funpart  F )  X.  { (/) } )
 )
 
Theorembrv 23825 The binary relationship over  _V always holds. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  A _V B
 
Theoremtxpss3v 23826 A tail cross product is a subset of the class of ordered triples. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  ( A  (x)  B )  C_  ( _V  X.  ( _V 
 X.  _V ) )
 
Theoremtxprel 23827 A tail cross product is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  Rel  ( A  (x)  B )
 
Theorembrtxp 23828 Characterize a trinary relationship over a tail cross product. Together with txpss3v 23826, this completely defines membership in a tail cross. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  X  e.  _V   &    |-  Y  e.  _V   &    |-  Z  e.  _V   =>    |-  ( X ( A 
 (x)  B ) <. Y ,  Z >. 
 <->  ( X A Y  /\  X B Z ) )
 
Theorembrtxp2 23829* The binary relationship over a tail cross when the second argument is not an ordered pair. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  A  e.  _V   =>    |-  ( A ( R 
 (x)  S ) B  <->  E. x E. y
 ( B  =  <. x ,  y >.  /\  A R x  /\  A S y ) )
 
Theoremdfpprod2 23830 Expanded definition of parallel product. (Contributed by Scott Fenton, 3-May-2014.)
 |- pprod ( A ,  B )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( A  o.  ( 1st  |`  ( _V 
 X.  _V ) ) ) )  i^i  ( `' ( 2nd  |`  ( _V 
 X.  _V ) )  o.  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) )
 
Theorempprodcnveq 23831 A converse law for parallel product. (Contributed by Scott Fenton, 3-May-2014.)
 |- pprod ( R ,  S )  =  `'pprod ( `' R ,  `' S )
 
Theorempprodss4v 23832 The parallel product is a subclass of  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) ). (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |- pprod ( A ,  B )  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
 
Theorembrpprod 23833 Characterize a quatary relationship over a tail cross product. Together with pprodss4v 23832, this completely defines membership in a parallel product. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  X  e.  _V   &    |-  Y  e.  _V   &    |-  Z  e.  _V   &    |-  W  e.  _V   =>    |-  ( <. X ,  Y >.pprod ( A ,  B )
 <. Z ,  W >.  <->  ( X A Z  /\  Y B W ) )
 
Theorembrpprod3a 23834* Condition for parallel product when the last argument is not an ordered pair. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  X  e.  _V   &    |-  Y  e.  _V   &    |-  Z  e.  _V   =>    |-  ( <. X ,  Y >.pprod ( R ,  S ) Z  <->  E. z E. w ( Z  =  <. z ,  w >.  /\  X R z  /\  Y S w ) )
 
Theorembrpprod3b 23835* Condition for parallel product when the first argument is not an ordered pair. (Contributed by Scott Fenton, 3-May-2014.)
 |-  X  e.  _V   &    |-  Y  e.  _V   &    |-  Z  e.  _V   =>    |-  ( Xpprod ( R ,  S ) <. Y ,  Z >.  <->  E. z E. w ( X  =  <. z ,  w >.  /\  z R Y  /\  w S Z ) )
 
Theoremrelsset 23836 The subset class is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  Rel  SSet
 
Theorembrsset 23837 For sets, the  SSet binary relationship is equivalent to the subset relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  B  e.  _V   =>    |-  ( A SSet B  <->  A 
 C_  B )
 
Theoremidsset 23838  _I is equal to  SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  _I  =  ( SSet  i^i  `' SSet )
 
Theoremeltrans 23839 Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  A  e.  _V   =>    |-  ( A  e.  Trans  <->  Tr  A )
 
Theoremdfon3 23840 A quantifier-free definition of  On. (Contributed by Scott Fenton, 5-Apr-2012.)
 |-  On  =  ( _V  \  ran  ( ( SSet  i^i  ( Trans  X.  _V ) ) 
 \  (  _I  u.  _E  ) ) )
 
Theoremdfon4 23841 Another quantifier-free definition of  On. (Contributed by Scott Fenton, 4-May-2014.)
 |-  On  =  ( _V  \  (
 ( SSet  \  (  _I 
 u.  _E  ) ) "
 Trans ) )
 
Theorembrtxpsd 23842* Expansion of a common form used in quantifier-free definitions. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( -.  A ran  ( ( _V  (x)  _E  )(++) ( R  (x)  _V )
 ) B  <->  A. x ( x  e.  B  <->  x R A ) )
 
Theorembrtxpsd2 23843* Another common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 21-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  R  =  ( C  \  ran  ( ( _V  (x)  _E  )(++) ( S  (x)  _V ) ) )   &    |-  A C B   =>    |-  ( A R B  <->  A. x ( x  e.  B  <->  x S A ) )
 
Theorembrtxpsd3 23844* A third common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 3-May-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  R  =  ( C  \  ran  ( ( _V  (x)  _E  )(++) ( S  (x)  _V ) ) )   &    |-  A C B   &    |-  ( x  e.  X  <->  x S A )   =>    |-  ( A R B  <->  B  =  X )
 
Theoremrelbigcup 23845 The  Bigcup relationship is a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  Rel  Bigcup
 
Theorembrbigcup 23846 Binary relationship over 
Bigcup. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  B  e.  _V   =>    |-  ( A Bigcup B  <->  U. A  =  B )
 
Theoremdfbigcup2 23847  Bigcup using maps-to notation. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Bigcup  =  ( x  e.  _V  |->  U. x )
 
Theoremfobigcup 23848  Bigcup maps the universe onto itself. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Bigcup : _V -onto-> _V
 
Theoremfnbigcup 23849  Bigcup is a function over the universal class. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  Bigcup  Fn  _V
 
Theoremfvbigcup 23850 For sets,  Bigcup yields union. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  A  e.  _V   =>    |-  ( Bigcup `  A )  =  U. A
 
Theoremelfix 23851 Membership in the fixpoints of a class. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  A  e.  _V   =>    |-  ( A  e.  Fix R  <->  A R A )
 
Theoremelfix2 23852 Alternative membership in the fixpoint of a class. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  Rel  R   =>    |-  ( A  e.  Fix R  <->  A R A )
 
Theoremdffix2 23853 The fixpoints of a class in terms of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Fix A  =  ran  (  A  i^i  _I  )
 
Theoremfixssdm 23854 The fixpoints of a class are a subset of its domain. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Fix A 
 C_  dom  A
 
Theoremfixssrn 23855 The fixpoints of a class are a subset of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Fix A 
 C_  ran  A
 
Theoremfixcnv 23856 The fixpoints of a class are the same as those of its converse. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Fix A  =  Fix `' A
 
Theoremfixun 23857 The fixpoint operator distributes over union. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Fix ( A  u.  B )  =  ( Fix A  u.  Fix B )
 
Theoremellimits 23858 Membership in the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  A  e.  _V   =>    |-  ( A  e.  Limits  <->  Lim  A )
 
Theoremlimitssson 23859 The class of all limit ordinals is a subclass of the class of all ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  Limits  C_  On
 
Theoremdfom5b 23860 A quantifier-free definition of 
om that does not depend on ax-inf 7334. (Note: label was changed from dfom5 7346 to dfom5b 23860 to prevent naming conflict. NM 12-Feb-2013) (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  om  =  ( On  i^i  |^| Limits )
 
Theoremelfuns 23861 Membership in the class of all functions. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  F  e.  _V   =>    |-  ( F  e.  Funs  <->  Fun  F )
 
Theoremelfunsg 23862 Closed form of elfuns 23861. (Contributed by Scott Fenton, 2-May-2014.)
 |-  ( F  e.  V  ->  ( F  e.  Funs  <->  Fun  F ) )
 
Theorembrsingle 23863 The binary relationship form of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ASingleton B  <->  B  =  { A } )
 
Theoremelsingles 23864* Membership in the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
 |-  ( A  e.  Singletons 
 <-> 
 E. x  A  =  { x } )
 
Theoremfnsingle 23865 The singleton relationship is a function over the universe. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |- Singleton  Fn  _V
 
Theoremfvsingle 23866 The value of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( A  e.  V  ->  (Singleton `  A )  =  { A } )
 
Theoremdfsingles2 23867* Alternate definition of the class of all singletons. (Contributed by Scott Fenton, 20-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  Singletons  =  { x  |  E. y  x  =  { y } }
 
Theoremsnelsingles 23868 A singleton is a member of the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
 |-  A  e.  _V   =>    |- 
 { A }  e.  Singletons
 
Theoremdfiota3 23869 A definiton of iota using minimal quantifiers. (Contributed by Scott Fenton, 19-Feb-2013.)
 |-  ( iota x ph )  = 
 U. U. ( { { x  |  ph } }  i^i 
 Singletons )
 
Theoremdffv4 23870 Another quantifier free definition of function value. (Contributed by Scott Fenton, 19-Feb-2013.)
 |-  ( F `  A )  = 
 U. U. ( { ( F " { A }
 ) }  i^i  Singletons )
 
Theoremunisnif 23871 Express union of singleton in terms of  if. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  U. { A }  =  if ( A  e.  _V ,  A ,  (/) )
 
Theorembrimage 23872 Binary relationship form of the Image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( AImage R B  <->  B  =  ( R " A ) )
 
Theorembrimageg 23873 Closed form of brimage 23872. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( AImage R B  <->  B  =  ( R " A ) ) )
 
Theoremfunimage 23874 Image A is a function. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  Fun Image A
 
Theoremfnimage 23875* Image R is a function over the set-like portion of  R. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |- Image R  Fn  { x  |  ( R
 " x )  e. 
 _V }
 
Theoremimageval 23876* The image functor in maps-to notation. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |- Image R  =  ( x  e.  _V  |->  ( R " x ) )
 
Theoremfvimage 23877 The value of the image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( A  e.  V  /\  ( R " A )  e.  W )  ->  (Image R `  A )  =  ( R " A ) )
 
Theorembrcart 23878 Binary relationship form of the cartesian product operator. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >.Cart C  <->  C  =  ( A  X.  B ) )
 
Theorembrdomain 23879 The binary relationship form of the domain function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ADomain B  <->  B  =  dom  A )
 
Theorembrrange 23880 The binary relationship form of the range function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ARange B  <->  B  =  ran  A )
 
Theorembrdomaing 23881 Closed form of brdomain 23879. (Contributed by Scott Fenton, 2-May-2014.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( ADomain B  <->  B  =  dom  A ) )
 
Theorembrrangeg 23882 Closed form of brrange 23880. (Contributed by Scott Fenton, 3-May-2014.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( ARange B  <->  B  =  ran  A ) )
 
Theorembrimg 23883 The binary relationship form of the Img function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >.Img C  <->  C  =  ( A " B ) )
 
Theorembrapply 23884 The binary relationship form of the Apply function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >.Apply C  <->  C  =  ( A `  B ) )
 
Theorembrcup 23885 Binary relationship form of the Cup function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >.Cup C  <->  C  =  ( A  u.  B ) )
 
Theorembrcap 23886 Binary relationship form of the Cap function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >.Cap C  <->  C  =  ( A  i^i  B ) )
 
Theorembrsuccf 23887 Binary relationship form of the Succ function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ASucc B  <->  B  =  suc  A )
 
Theoremfunpartfun 23888 The functional part of  F is a function. (Contributed by Scott Fenton, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  Fun Funpart F
 
Theoremfunpartss 23889 The functional part of  F is a subset of  F. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |- Funpart F  C_  F
 
Theoremfunpartfv 23890 The function value of the functional part is identical to the original functional value. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (Funpart F `
  A )  =  ( F `  A )
 
Theoremfullfunfnv 23891 The full functional part of  F is a function over  _V. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |- FullFun F  Fn  _V
 
Theoremfullfunfv 23892 The function value of the full function of  F agrees with  F. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (FullFun F `
  A )  =  ( F `  A )
 
Theorembrfullfun 23893 A binary relationship form condition for the full function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( AFullFun F B  <->  B  =  ( F `  A ) )
 
Theorembrrestrict 23894 The binary relationship form of the Restrict function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >.Restrict
 C 
 <->  C  =  ( A  |`  B ) )
 
Theoremdfrdg4 23895 A quantifier-free definition of the recursive definition generator. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  rec ( F ,  A )  =  U. ( (
 Funs  i^i  ( `'Domain " On ) )  \  dom  (
 ( `'  _E  o. Domain ) 
 \  Fix ( `'Apply  o.  (
 ( ( _V  X.  { (/) } )  X.  { U. { A } }
 )  u.  ( ( ( Bigcup  o. Img )  |`  ( _V 
 X.  Limits ) )  u.  ( (FullFun F  o.  (Apply  o. pprod (  _I  ,  Bigcup ) ) )  |`  ( _V  X.  ran Succ ) ) ) ) ) ) )
 
Theoremtfrqfree 23896* Calculate a quantifier-free version of the function from tfr1 6408 through tfr3 6410. (Contributed by Scott Fenton, 29-Apr-2014.)
 |-  (
 ( Funs  i^i  ( `'Domain " On ) )  \  dom  ( ( `'  _E  o. Domain )  \  Fix ( `'Apply  o.  (FullFun G  o. Restrict ) ) ) )  =  { f  |  E. x  e.  On  (
 f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }
 
18.7.29  Alternate ordered pairs
 
Syntaxcaltop 23897 Declare the syntax for an alternate ordered pair.
 class  << A ,  B >>
 
Syntaxcaltxp 23898 Declare the syntax for an alternate cross product.
 class  ( A 
 XX.  B )
 
Definitiondf-altop 23899 An alternative definition of ordered pairs. This definition removes a hypothesis from its defining theorem (see altopth 23910), making it more convenient in some circumstances. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  << A ,  B >>  =  { { A } ,  { A ,  { B } } }
 
Definitiondf-altxp 23900* Define cross products of alternative ordered pairs. (Contributed by Scott Fenton, 23-Mar-2012.)
 |-  ( A  XX.  B )  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  << x ,  y >> }
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