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Theorem List for Metamath Proof Explorer - 25401-25500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxccart 25401 Declare the syntax for the cartesian function.
 class Cart
 
Syntaxcimg 25402 Declare the syntax for the image function.
 class Img
 
Syntaxcdomain 25403 Declare the syntax for the domain function.
 class Domain
 
Syntaxcrange 25404 Declare the syntax for the range function.
 class Range
 
Syntaxcapply 25405 Declare the syntax for the application function.
 class Apply
 
Syntaxccup 25406 Declare the syntax for the cup function.
 class Cup
 
Syntaxccap 25407 Declare the syntax for the cap function.
 class Cap
 
Syntaxcsuccf 25408 Declare the syntax for the successor function.
 class Succ
 
Syntaxcfunpart 25409 Declare the syntax for the functional part functor.
 class Funpart F
 
Syntaxcfullfn 25410 Declare the syntax for the full function functor.
 class FullFun F
 
Syntaxcrestrict 25411 Declare the syntax for the restriction function.
 class Restrict
 
Definitiondf-txp 25412 Define the tail cross of two classes. Membership in this class is defined by txpss3v 25435 and brtxp 25437. (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 25413 Define the parallel product of two classes. Membership in this class is defined by pprodss4v 25441 and brpprod 25442. (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 25414 Define the subset class. For the value, see brsset 25446. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  SSet  =  ( ( _V  X.  _V )  \  ran  (  _E  (x)  ( _V  \  _E  ) ) )
 
Definitiondf-trans 25415 Define the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  Trans  =  ( _V  \  ran  (
 (  _E  o.  _E  )  \  _E  ) )
 
Definitiondf-bigcup 25416 Define the Bigcup function, which, per fvbigcup 25459, 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 25417 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 25418 Define the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  Limits  =  ( ( On  i^i  Fix Bigcup )  \  { (/) } )
 
Definitiondf-funs 25419 Define the class of all functions. See elfuns 25471 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 25420 Define the singleton function. See brsingle 25473 for its value. (Contributed by Scott Fenton, 4-Apr-2014.)
 |- Singleton  =  ( ( _V  X.  _V )  \  ran  ( ( _V  (x)  _E  )(++) (  _I  (x)  _V )
 ) )
 
Definitiondf-singles 25421 Define the class of all singletons. See elsingles 25474 for membership. (Contributed by Scott Fenton, 19-Feb-2013.)
 |-  Singletons  =  ran Singleton
 
Definitiondf-image 25422 Define the image functor. This function takes a set  A to a function  x  |->  ( A
" x ), providing that the latter exists. See imageval 25486 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 25423 Define the cartesian product function. See brcart 25488 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 25424 Define the image function. See brimg 25493 for its value. (Contributed by Scott Fenton, 12-Apr-2014.)
 |- Img  =  (Image ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) )  o. Cart
 )
 
Definitiondf-domain 25425 Define the domain function. See brdomain 25489 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
 |- Domain  = Image ( 1st  |`  ( _V  X.  _V ) )
 
Definitiondf-range 25426 Define the range function. See brrange 25490 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
 |- Range  = Image ( 2nd  |`  ( _V  X.  _V ) )
 
Definitiondf-cup 25427 Define the little cup function. See brcup 25495 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 25428 Define the little cap function. See brcap 25496 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 25429 Define the restriction function. See brrestrict 25505 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
 |- Restrict  =  (Cap 
 o.  ( 1st  (x)  (Cart  o.  ( 2nd  (x)  (Range  o. 
 1st ) ) ) ) )
 
Definitiondf-succf 25430 Define the successor function. See brsuccf 25497 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
 |- Succ  =  (Cup 
 o.  (  _I  (x) Singleton ) )
 
Definitiondf-apply 25431 Define the application function. See brapply 25494 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 25432 Define the functional part of a class  F. This is the maximal part of  F that is a function. See funpartfun 25499 and funpartfv 25501 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 25433 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 25434 The binary relationship over  _V always holds. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  A _V B
 
Theoremtxpss3v 25435 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 25436 A tail cross product is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  Rel  ( A  (x)  B )
 
Theorembrtxp 25437 Characterize a trinary relationship over a tail cross product. Together with txpss3v 25435, 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 25438* 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 25439 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 25440 A converse law for parallel product. (Contributed by Scott Fenton, 3-May-2014.)
 |- pprod ( R ,  S )  =  `'pprod ( `' R ,  `' S )
 
Theorempprodss4v 25441 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 25442 Characterize a quatary relationship over a tail cross product. Together with pprodss4v 25441, 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 25443* 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 25444* 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 25445 The subset class is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  Rel  SSet
 
Theorembrsset 25446 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 25447  _I is equal to  SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  _I  =  ( SSet  i^i  `' SSet )
 
Theoremeltrans 25448 Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  A  e.  _V   =>    |-  ( A  e.  Trans  <->  Tr  A )
 
Theoremdfon3 25449 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 25450 Another quantifier-free definition of  On. (Contributed by Scott Fenton, 4-May-2014.)
 |-  On  =  ( _V  \  (
 ( SSet  \  (  _I 
 u.  _E  ) ) "
 Trans ) )
 
Theorembrtxpsd 25451* 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 25452* 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 25453* 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 25454 The  Bigcup relationship is a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  Rel  Bigcup
 
Theorembrbigcup 25455 Binary relationship over 
Bigcup. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  B  e.  _V   =>    |-  ( A Bigcup B  <->  U. A  =  B )
 
Theoremdfbigcup2 25456  Bigcup using maps-to notation. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Bigcup  =  ( x  e.  _V  |->  U. x )
 
Theoremfobigcup 25457  Bigcup maps the universe onto itself. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Bigcup : _V -onto-> _V
 
Theoremfnbigcup 25458  Bigcup is a function over the universal class. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  Bigcup  Fn  _V
 
Theoremfvbigcup 25459 For sets,  Bigcup yields union. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  A  e.  _V   =>    |-  ( Bigcup `  A )  =  U. A
 
Theoremelfix 25460 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 25461 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 25462 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 25463 The fixpoints of a class are a subset of its domain. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Fix A 
 C_  dom  A
 
Theoremfixssrn 25464 The fixpoints of a class are a subset of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Fix A 
 C_  ran  A
 
Theoremfixcnv 25465 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 25466 The fixpoint operator distributes over union. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Fix ( A  u.  B )  =  ( Fix A  u.  Fix B )
 
Theoremellimits 25467 Membership in the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  A  e.  _V   =>    |-  ( A  e.  Limits  <->  Lim  A )
 
Theoremlimitssson 25468 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 25469 A quantifier-free definition of 
om that does not depend on ax-inf 7519. (Note: label was changed from dfom5 7531 to dfom5b 25469 to prevent naming conflict. NM 12-Feb-2013) (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  om  =  ( On  i^i  |^| Limits )
 
Theoremdffun10 25470 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 25471 Membership in the class of all functions. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  F  e.  _V   =>    |-  ( F  e.  Funs  <->  Fun  F )
 
Theoremelfunsg 25472 Closed form of elfuns 25471. (Contributed by Scott Fenton, 2-May-2014.)
 |-  ( F  e.  V  ->  ( F  e.  Funs  <->  Fun  F ) )
 
Theorembrsingle 25473 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 25474* Membership in the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
 |-  ( A  e.  Singletons 
 <-> 
 E. x  A  =  { x } )
 
Theoremfnsingle 25475 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 25476 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 25477* 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 25478 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 25479 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 25480 Another quantifier free definition of function value. (Contributed by Scott Fenton, 19-Feb-2013.)
 |-  ( F `  A )  = 
 U. U. ( { ( F " { A }
 ) }  i^i  Singletons )
 
Theoremunisnif 25481 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 25482 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 25483 Closed form of brimage 25482. (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 25484 Image A is a function. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  Fun Image A
 
Theoremfnimage 25485* 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 25486* 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 25487 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 25488 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 25489 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 25490 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 25491 Closed form of brdomain 25489. (Contributed by Scott Fenton, 2-May-2014.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( ADomain B  <->  B  =  dom  A ) )
 
Theorembrrangeg 25492 Closed form of brrange 25490. (Contributed by Scott Fenton, 3-May-2014.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( ARange B  <->  B  =  ran  A ) )
 
Theorembrimg 25493 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 25494 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 25495 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 25496 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 25497 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 )
 
Theoremfunpartlem 25498* Lemma for funpartfun 25499. Show membership in the restriction. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  ( A  e.  dom  ( (Image
 F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E. x ( F
 " { A }
 )  =  { x } )
 
Theoremfunpartfun 25499 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 25500 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
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