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Theorem List for Metamath Proof Explorer - 24401-24500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-limits 24401 Define the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  Limits  =  ( ( On  i^i  Fix Bigcup )  \  { (/) } )
 
Definitiondf-funs 24402 Define the class of all functions. See elfuns 24454 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 24403 Define the singleton function. See brsingle 24456 for its value. (Contributed by Scott Fenton, 4-Apr-2014.)
 |- Singleton  =  ( ( _V  X.  _V )  \  ran  ( ( _V  (x)  _E  )(++) (  _I  (x)  _V )
 ) )
 
Definitiondf-singles 24404 Define the class of all singletons. See elsingles 24457 for membership. (Contributed by Scott Fenton, 19-Feb-2013.)
 |-  Singletons  =  ran Singleton
 
Definitiondf-image 24405 Define the image functor. This function takes a set  A to a function  x  |->  ( A
" x ), providing that the latter exists. See imageval 24469 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 24406 Define the cartesian product function. See brcart 24471 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 24407 Define the image function. See brimg 24476 for its value. (Contributed by Scott Fenton, 12-Apr-2014.)
 |- Img  =  (Image ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) )  o. Cart
 )
 
Definitiondf-domain 24408 Define the domain function. See brdomain 24472 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
 |- Domain  = Image ( 1st  |`  ( _V  X.  _V ) )
 
Definitiondf-range 24409 Define the range function. See brrange 24473 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
 |- Range  = Image ( 2nd  |`  ( _V  X.  _V ) )
 
Definitiondf-cup 24410 Define the little cup function. See brcup 24478 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 24411 Define the little cap function. See brcap 24479 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 24412 Define the restriction function. See brrestrict 24487 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
 |- Restrict  =  (Cap 
 o.  ( 1st  (x)  (Cart  o.  ( 2nd  (x)  (Range  o. 
 1st ) ) ) ) )
 
Definitiondf-succf 24413 Define the successor function. See brsuccf 24480 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
 |- Succ  =  (Cup 
 o.  (  _I  (x) Singleton ) )
 
Definitiondf-apply 24414 Define the application function. See brapply 24477 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 24415 Define the functional part of a class  F. This is the maximal part of  F that is a function. See funpartfun 24481 and funpartfv 24483 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 24416 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 24417 The binary relationship over  _V always holds. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  A _V B
 
Theoremtxpss3v 24418 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 24419 A tail cross product is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  Rel  ( A  (x)  B )
 
Theorembrtxp 24420 Characterize a trinary relationship over a tail cross product. Together with txpss3v 24418, 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 24421* 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 24422 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 24423 A converse law for parallel product. (Contributed by Scott Fenton, 3-May-2014.)
 |- pprod ( R ,  S )  =  `'pprod ( `' R ,  `' S )
 
Theorempprodss4v 24424 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 24425 Characterize a quatary relationship over a tail cross product. Together with pprodss4v 24424, 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 24426* 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 24427* 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 24428 The subset class is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  Rel  SSet
 
Theorembrsset 24429 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 24430  _I is equal to  SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  _I  =  ( SSet  i^i  `' SSet )
 
Theoremeltrans 24431 Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  A  e.  _V   =>    |-  ( A  e.  Trans  <->  Tr  A )
 
Theoremdfon3 24432 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 24433 Another quantifier-free definition of  On. (Contributed by Scott Fenton, 4-May-2014.)
 |-  On  =  ( _V  \  (
 ( SSet  \  (  _I 
 u.  _E  ) ) "
 Trans ) )
 
Theorembrtxpsd 24434* 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 24435* 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 24436* 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 24437 The  Bigcup relationship is a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  Rel  Bigcup
 
Theorembrbigcup 24438 Binary relationship over 
Bigcup. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  B  e.  _V   =>    |-  ( A Bigcup B  <->  U. A  =  B )
 
Theoremdfbigcup2 24439  Bigcup using maps-to notation. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Bigcup  =  ( x  e.  _V  |->  U. x )
 
Theoremfobigcup 24440  Bigcup maps the universe onto itself. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Bigcup : _V -onto-> _V
 
Theoremfnbigcup 24441  Bigcup is a function over the universal class. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  Bigcup  Fn  _V
 
Theoremfvbigcup 24442 For sets,  Bigcup yields union. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  A  e.  _V   =>    |-  ( Bigcup `  A )  =  U. A
 
Theoremelfix 24443 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 24444 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 24445 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 24446 The fixpoints of a class are a subset of its domain. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Fix A 
 C_  dom  A
 
Theoremfixssrn 24447 The fixpoints of a class are a subset of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Fix A 
 C_  ran  A
 
Theoremfixcnv 24448 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 24449 The fixpoint operator distributes over union. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Fix ( A  u.  B )  =  ( Fix A  u.  Fix B )
 
Theoremellimits 24450 Membership in the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  A  e.  _V   =>    |-  ( A  e.  Limits  <->  Lim  A )
 
Theoremlimitssson 24451 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 24452 A quantifier-free definition of 
om that does not depend on ax-inf 7339. (Note: label was changed from dfom5 7351 to dfom5b 24452 to prevent naming conflict. NM 12-Feb-2013) (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  om  =  ( On  i^i  |^| Limits )
 
Theoremdffun10 24453 Another potential definition of functionhood. Based on statements in http://people.math.gatech.edu/~belinfan/research/autoreas/otter/sum/fs/. (Contributed by Scott Fenton, 30-Aug-2017.)
 |-  ( Fun  F  <->  F  C_  (  _I 
 o.  ( _V  \  (
 ( _V  \  _I  )  o.  F ) ) ) )
 
Theoremelfuns 24454 Membership in the class of all functions. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  F  e.  _V   =>    |-  ( F  e.  Funs  <->  Fun  F )
 
Theoremelfunsg 24455 Closed form of elfuns 24454. (Contributed by Scott Fenton, 2-May-2014.)
 |-  ( F  e.  V  ->  ( F  e.  Funs  <->  Fun  F ) )
 
Theorembrsingle 24456 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 24457* Membership in the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
 |-  ( A  e.  Singletons 
 <-> 
 E. x  A  =  { x } )
 
Theoremfnsingle 24458 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 24459 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 24460* 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 24461 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 24462 A definiton of iota using minimal quantifiers. (Contributed by Scott Fenton, 19-Feb-2013.)
 |-  ( iota x ph )  = 
 U. U. ( { { x  |  ph } }  i^i 
 Singletons )
 
Theoremdffv5 24463 Another quantifier free definition of function value. (Contributed by Scott Fenton, 19-Feb-2013.)
 |-  ( F `  A )  = 
 U. U. ( { ( F " { A }
 ) }  i^i  Singletons )
 
Theoremunisnif 24464 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 24465 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 24466 Closed form of brimage 24465. (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 24467 Image A is a function. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  Fun Image A
 
Theoremfnimage 24468* 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 24469* 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 24470 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 24471 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 24472 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 24473 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 24474 Closed form of brdomain 24472. (Contributed by Scott Fenton, 2-May-2014.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( ADomain B  <->  B  =  dom  A ) )
 
Theorembrrangeg 24475 Closed form of brrange 24473. (Contributed by Scott Fenton, 3-May-2014.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( ARange B  <->  B  =  ran  A ) )
 
Theorembrimg 24476 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 24477 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 24478 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 24479 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 24480 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 24481 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 24482 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 24483 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 24484 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 24485 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 24486 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 24487 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 24488 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 24489* Calculate a quantifier-free version of the function from tfr1 6413 through tfr3 6415. (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 ) ) ) }
 
18.7.31  Alternate ordered pairs
 
Syntaxcaltop 24490 Declare the syntax for an alternate ordered pair.
 class  << A ,  B >>
 
Syntaxcaltxp 24491 Declare the syntax for an alternate cross product.
 class  ( A 
 XX.  B )
 
Definitiondf-altop 24492 An alternative definition of ordered pairs. This definition removes a hypothesis from its defining theorem (see altopth 24503), making it more convenient in some circumstances. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  << A ,  B >>  =  { { A } ,  { A ,  { B } } }
 
Definitiondf-altxp 24493* 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 >> }
 
Theoremaltopex 24494 Alternative ordered pairs always exist. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  << A ,  B >>  e.  _V
 
Theoremaltopthsn 24495 Two alternate ordered pairs are equal iff the singletons of their respective elements are equal. Note that this holds regardless of sethood of any of the elements. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  ( <<
 A ,  B >>  = 
 << C ,  D >>  <->  ( { A }  =  { C }  /\  { B }  =  { D } )
 )
 
Theoremaltopeq12 24496 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  (
 ( A  =  B  /\  C  =  D ) 
 ->  << A ,  C >> 
 =  << B ,  D >> )
 
Theoremaltopeq1 24497 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  ( A  =  B  ->  << A ,  C >>  =  << B ,  C >> )
 
Theoremaltopeq2 24498 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  ( A  =  B  ->  << C ,  A >>  =  << C ,  B >> )
 
Theoremaltopth1 24499 Equality of the first members of equal alternate ordered pairs, which holds regardless of the second members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  ( A  e.  V  ->  (
 << A ,  B >>  = 
 << C ,  D >>  ->  A  =  C )
 )
 
Theoremaltopth2 24500 Equality of the second members of equal alternate ordered pairs, which holds regardless of the first members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  ( B  e.  V  ->  (
 << A ,  B >>  = 
 << C ,  D >>  ->  B  =  D )
 )
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