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Theorem List for Metamath Proof Explorer - 23501-23600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembdayelon 23501 The value of the birthday function is always an ordinal. (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  ( bday `  A )  e. 
 On
 
Theoremnoprc 23502 The surreal numbers are a proper class. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  -.  No  e.  _V
 
16.6.25  Surreal Numbers: Density
 
Theoremaxdenselem1 23503 Lemma for axdense 23511. A surreal is a function over its birthday. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  ( A  e.  No  ->  A  Fn  ( bday `  A ) )
 
Theoremaxdenselem2 23504 Lemma for axfe (future) and axdense 23511. The value of a surreal at its birthday is  (/). (Shortened proof on 2012-Apr-14, SF) (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  ( A  e.  No  ->  ( A `  ( bday `  A ) )  =  (/) )
 
Theoremaxdenselem3 23505* Lemma for axdense 23511. If one surreal is older than another, then there is an ordinal at which they are not equal. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  (
 ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A )  e.  ( bday `  B )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a ) ) )
 
Theoremaxdenselem4 23506* Lemma for axdense 23511. Show that a particular abstraction is an ordinal. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  (
 ( ( A  e.  No  /\  B  e.  No )  /\  A < s B )  ->  |^| { a  e.  On  |  ( A `
  a )  =/=  ( B `  a
 ) }  e.  On )
 
Theoremaxdenselem5 23507* Lemma for axdense 23511. If the birthdays of two distinct surreals are equal, then the ordinal from axdenselem4 23506 is an element of that birthday. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  (
 ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
 bday `  B )  /\  A < s B ) )  ->  |^| { a  e.  On  |  ( A `
  a )  =/=  ( B `  a
 ) }  e.  ( bday `  A ) )
 
Theoremaxdenselem6 23508* The restriction of a surreal to the abstraction from axdenselem4 23506 is still a surreal. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  (
 ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
 bday `  B )  /\  A < s B ) )  ->  ( A  |` 
 |^| { a  e.  On  |  ( A `  a
 )  =/=  ( B `  a ) } )  e.  No )
 
Theoremaxdenselem7 23509* Lemma for axdense 23511. 
A and  B are equal at all elements of the abstraction. (Contributed by Scott Fenton, 17-Jun-2011.)
 |-  (
 ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
 bday `  B )  /\  A < s B ) )  ->  ( C  e.  |^| { a  e. 
 On  |  ( A `
  a )  =/=  ( B `  a
 ) }  ->  ( A `  C )  =  ( B `  C ) ) )
 
Theoremaxdenselem8 23510* Lemma for axdense 23511. Give a condition for surreal less than when two surreals have the same birthday. (Contributed by Scott Fenton, 19-Jun-2011.)
 |-  (
 ( A  e.  No  /\  B  e.  No  /\  ( bday `  A )  =  ( bday `  B )
 )  ->  ( A < s B  <->  ( ( A `
  |^| { a  e. 
 On  |  ( A `
  a )  =/=  ( B `  a
 ) } )  =  1o  /\  ( B `
  |^| { a  e. 
 On  |  ( A `
  a )  =/=  ( B `  a
 ) } )  =  2o ) ) )
 
Theoremaxdense 23511* Given two distinct surreals with the same birthday, there is an older surreal lying between the two of them. Alling's axiom (SD) (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  (
 ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
 bday `  B )  /\  A < s B ) )  ->  E. x  e.  No  ( ( bday `  x )  e.  ( bday `  A )  /\  A < s x  /\  x < s B ) )
 
Theoremnocvxminlem 23512* Lemma for nocvxmin 23513. Given two birthday-minimal elements of a convex class of surreals, they are not comparable. (Contributed by Scott Fenton, 30-Jun-2011.)
 |-  (
 ( A  C_  No  /\ 
 A. x  e.  A  A. y  e.  A  A. z  e.  No  (
 ( x < s
 z  /\  z < s y )  ->  z  e.  A ) )  ->  ( ( ( X  e.  A  /\  Y  e.  A )  /\  (
 ( bday `  X )  =  |^| ( bday " A )  /\  ( bday `  Y )  =  |^| ( bday " A ) ) ) 
 ->  -.  X < s Y ) )
 
Theoremnocvxmin 23513* Given a non-empty convex class of surreals, there is a unique birthday-minimal element of that class. (Contributed by Scott Fenton, 30-Jun-2011.)
 |-  (
 ( A  =/=  (/)  /\  A  C_ 
 No  /\  A. x  e.  A  A. y  e.  A  A. z  e. 
 No  ( ( x < s z  /\  z < s y ) 
 ->  z  e.  A ) )  ->  E! w  e.  A  ( bday `  w )  =  |^| ( bday " A ) )
 
16.6.26  Surreal Numbers: Full-Eta Property
 
Theoremaxfelem1 23514 Lemma for axfe (future) . The successor of the union of the image of the birthday function under a set is an ordinal. (Contributed by Scott Fenton, 20-Aug-2011.)
 |-  ( A  e.  V  ->  suc  U. ( bday " A )  e. 
 On )
 
Theoremaxfelem2 23515* Lemma for axfe (future) . Show a particular abstraction is an ordinal. (Contributed by Scott Fenton, 17-Aug-2011.)
 |-  X  e.  { 1o ,  2o }   &    |-  C  =  |^| { a  e.  On  |  A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  X }   =>    |-  (
 ( F  C_  No  /\  F  e.  A ) 
 ->  C  e.  On )
 
Theoremaxfelem3 23516* Lemma for axfe (future) . Calculate the birthday of  ( C  X.  { X } ). (Contributed by Scott Fenton, 17-Aug-2011.)
 |-  X  e.  { 1o ,  2o }   &    |-  C  =  |^| { a  e.  On  |  A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  X }   =>    |-  (
 ( F  C_  No  /\  F  e.  A ) 
 ->  ( bday `  ( C  X.  { X } )
 )  =  C )
 
Theoremaxfelem4 23517* Lemma for axfe (future) . The infimum of the class of all ordinals such that  A is not  X is an ordinal. (Contributed by Scott Fenton, 17-Aug-2011.)
 |-  X  e.  { 1o ,  2o }   =>    |-  ( A  e.  No  -> 
 |^| { x  e.  On  |  ( A `  x )  =/=  X }  e.  On )
 
Theoremaxfelem5 23518* Lemma for axfe (future) . There is always a minimal value of a surreal that is not a given sign. (Contributed by Scott Fenton, 3-Aug-2011.)
 |-  X  e.  { 1o ,  2o }   =>    |-  ( A  e.  No  ->  ( A `  |^| { x  e.  On  |  ( A `
  x )  =/= 
 X } )  =/= 
 X )
 
Theoremaxfelem6 23519* Lemma for axfe (future) . Given an element  A of  F, then the first position where it differs from  X is strictly less than  C (Contributed by Scott Fenton, 3-Aug-2011.)
 |-  X  e.  { 1o ,  2o }   &    |-  C  =  |^| { a  e.  On  |  A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  X }   =>    |-  (
 ( F  C_  No  /\  A  e.  F ) 
 ->  |^| { x  e. 
 On  |  ( A `
  x )  =/= 
 X }  e.  C )
 
Theoremaxfelem7 23520* Lemma for axfe (future) . Calculate the value of  ( C  X.  { X } ) at the minimal ordinal whose value is different from  X. (Contributed by Scott Fenton, 3-Aug-2011.)
 |-  X  e.  { 1o ,  2o }   &    |-  C  =  |^| { a  e.  On  |  A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  X }   =>    |-  (
 ( F  C_  No  /\  A  e.  F ) 
 ->  ( ( C  X.  { X } ) `  |^|
 { x  e.  On  |  ( A `  x )  =/=  X } )  =  X )
 
Theoremaxfelem8 23521* Lemma for axfe (future) . Bound the birthday of  ( C  X.  { X } ) above. (Contributed by Scott Fenton, 17-Aug-2011.)
 |-  S  e.  { 1o ,  2o }   &    |-  C  =  |^| { a  e.  On  |  A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  S }   =>    |-  (
 ( ( F  C_  No  /\  F  e.  A )  /\  ( X  e.  On  /\  A. n  e.  F  ( bday `  n )  e.  X )
 )  ->  ( bday `  ( C  X.  { S } ) )  C_  X )
 
Theoremaxfelem9 23522* Lemma for axfe (future) . The full statement of the axiom when  R is empty. (Contributed by Scott Fenton, 3-Aug-2011.)
 |-  ( R  =  (/)  ->  (
 ( ( ( L 
 C_  No  /\  R  C_  No )  /\  ( L  e.  A  /\  R  e.  B ) )  /\  ( X  e.  On  /\ 
 A. n  e.  ( L  u.  R ) (
 bday `  n )  e.  X )  /\  A. x  e.  L  A. y  e.  R  x < s
 y )  ->  E. z  e.  No  ( ( bday `  z )  C_  X  /\  A. x  e.  L  x < s z  /\  A. y  e.  R  z < s y ) ) )
 
Theoremaxfelem10 23523* Lemma for axfe (future) . The full statement of the axiom when  L is empty. (Contributed by Scott Fenton, 3-Aug-2011.)
 |-  ( L  =  (/)  ->  (
 ( ( ( L 
 C_  No  /\  R  C_  No )  /\  ( L  e.  A  /\  R  e.  B ) )  /\  ( X  e.  On  /\ 
 A. n  e.  ( L  u.  R ) (
 bday `  n )  e.  X )  /\  A. x  e.  L  A. y  e.  R  x < s
 y )  ->  E. z  e.  No  ( ( bday `  z )  C_  X  /\  A. x  e.  L  x < s z  /\  A. y  e.  R  z < s y ) ) )
 
Theoremaxfelem11 23524* Lemma for axfe (future) . We are now going to construct a new surreal for the case where  L and  R are not null. The birthday/domain of this surreal will be the intersection of  D. This lemma just changes bound variables for later use. (Contributed by Scott Fenton, 22-Apr-2012.)
 |-  D  =  { a  e.  On  |  A. p  e.  L  A. q  e.  R  ( p  |`  a )  =/=  ( q  |`  a ) }   =>    |-  D  =  { b  e.  On  |  A. n  e.  L  A. m  e.  R  ( n  |`  b )  =/=  ( m  |`  b ) }
 
Theoremaxfelem12 23525* Lemma for axfe (future) . The surreal mentioned in the previous lemma will ultimately be the union of  C. Again, we just change bound variables for later. (Contributed by Scott Fenton, 22-Apr-2012.)
 |-  C  =  { f  |  E. g  e.  L  E. h  e.  R  E. a  e.  P  ( ( g  |`  a )  =  f 
 /\  ( h  |`  a )  =  f
 ) }   =>    |-  C  =  { b  |  E. c  e.  L  E. d  e.  R  E. e  e.  P  ( ( c  |`  e )  =  b  /\  ( d  |`  e )  =  b ) }
 
Theoremaxfelem13 23526 Lemma for axfe (future) . Establish an equality result for restriction of surreals. (Contributed by Scott Fenton, 22-Apr-2012.)
 |-  (
 ( A  C_  No  /\  X  e.  A ) 
 ->  ( X  |`  suc  U. ( bday " ( A  u.  B ) ) )  =  X )
 
Theoremaxfelem14 23527* Lemma for axfe (future) . Now, we prove that  P, the birthday of a surreal we are going to be working with, is actually an ordinal. (Contributed by Scott Fenton, 22-Apr-2012.)
 |-  D  =  { a  e.  On  |  A. p  e.  L  A. q  e.  R  ( p  |`  a )  =/=  ( q  |`  a ) }   &    |-  P  =  |^| D   =>    |-  ( ( ( ( L  C_  No  /\  R  C_ 
 No )  /\  ( L  e.  A  /\  R  e.  B )
 )  /\  A. x  e.  L  A. y  e.  R  x < s
 y )  ->  P  e.  On )
 
Theoremaxfelem15 23528* Lemma for axfe (future) . For non-empty  L and  R,  P is non-empty. (Contributed by Scott Fenton, 22-Apr-2012.)
 |-  D  =  { a  e.  On  |  A. p  e.  L  A. q  e.  R  ( p  |`  a )  =/=  ( q  |`  a ) }   &    |-  P  =  |^| D   =>    |-  ( ( L  =/=  (/)  /\  R  =/=  (/) )  ->  P  =/=  (/) )
 
Theoremaxfelem16 23529* Lemma for axfe (future) . Preliminary work for axfelem17 23530. (Contributed by Scott Fenton, 22-Apr-2012.)
 |-  (
 ( ( ( A 
 C_  No  /\  B  C_  No )  /\  A. x  e.  A  A. y  e.  B  x < s
 y )  /\  ( X  e.  On  /\  (
 ( P  e.  A  /\  Q  e.  B ) 
 /\  ( R  e.  A  /\  S  e.  B ) ) )  /\  ( ( P  |`  X )  =  ( Q  |`  X ) 
 /\  ( R  |`  X )  =  ( S  |`  X ) ) )  ->  ( -.  ( P  |`  X ) < s ( R  |`  X )  /\  -.  ( Q  |`  X ) < s ( S  |`  X ) ) )
 
Theoremaxfelem17 23530* Lemma for axfe (future) . A uniqueness result for restrictions of surreals. (Contributed by Scott Fenton, 22-Apr-2012.)
 |-  (
 ( ( ( A 
 C_  No  /\  B  C_  No )  /\  A. x  e.  A  A. y  e.  B  x < s
 y )  /\  ( X  e.  On  /\  (
 ( P  e.  A  /\  Q  e.  B ) 
 /\  ( R  e.  A  /\  S  e.  B ) ) )  /\  ( ( P  |`  X )  =  ( Q  |`  X ) 
 /\  ( R  |`  X )  =  ( S  |`  X ) ) )  ->  (
 ( P  |`  X )  =  ( R  |`  X ) 
 /\  ( Q  |`  X )  =  ( S  |`  X ) ) )
 
Theoremaxfelem18 23531* Lemma for axfe (future) . At this point, we introduce a new surreal number  M (for "middle"). The next several lemmas prove that either  M or a closely related surreal has the required properties for the final theorem. We begin by calculating the domain of  M. (Contributed by Scott Fenton, 22-Apr-2012.)
 |-  D  =  { a  e.  On  |  A. p  e.  L  A. q  e.  R  ( p  |`  a )  =/=  ( q  |`  a ) }   &    |-  P  =  |^| D   &    |-  C  =  { f  |  E. g  e.  L  E. h  e.  R  E. a  e.  P  ( ( g  |`  a )  =  f  /\  ( h  |`  a )  =  f ) }   &    |-  M  =  U. C   =>    |-  ( ( ( ( L  C_  No  /\  R  C_ 
 No )  /\  ( L  e.  A  /\  R  e.  B )
 )  /\  A. x  e.  L  A. y  e.  R  x < s
 y )  ->  dom  M  =  U. P )
 
Theoremaxfelem19 23532* Lemma for axfe (future) .  C is a subset of the surreals. (Contributed by Scott Fenton, 22-Apr-2012.)
 |-  D  =  { a  e.  On  |  A. p  e.  L  A. q  e.  R  ( p  |`  a )  =/=  ( q  |`  a ) }   &    |-  P  =  |^| D   &    |-  C  =  { f  |  E. g  e.  L  E. h  e.  R  E. a  e.  P  ( ( g  |`  a )  =  f  /\  ( h  |`  a )  =  f ) }   &    |-  M  =  U. C   =>    |-  ( ( ( ( L  C_  No  /\  R  C_ 
 No )  /\  ( L  e.  A  /\  R  e.  B )
 )  /\  A. x  e.  L  A. y  e.  R  x < s
 y )  ->  C  C_ 
 No )
 
Theoremaxfelem20 23533* Lemma for axfe (future) . Next, we prove that  M is a function. (Contributed by Scott Fenton, 22-Apr-2012.)
 |-  D  =  { a  e.  On  |  A. p  e.  L  A. q  e.  R  ( p  |`  a )  =/=  ( q  |`  a ) }   &    |-  P  =  |^| D   &    |-  C  =  { f  |  E. g  e.  L  E. h  e.  R  E. a  e.  P  ( ( g  |`  a )  =  f  /\  ( h  |`  a )  =  f ) }   &    |-  M  =  U. C   =>    |-  ( ( ( ( L  C_  No  /\  R  C_ 
 No )  /\  ( L  e.  A  /\  R  e.  B )
 )  /\  A. x  e.  L  A. y  e.  R  x < s
 y )  ->  Fun  M )
 
Theoremaxfelem21 23534* Lemma for axfe (future) . Now, using the previous three lemmas, we show that  M is a surreal. (Contributed by Scott Fenton, 22-Apr-2012.)
 |-  D  =  { a  e.  On  |  A. p  e.  L  A. q  e.  R  ( p  |`  a )  =/=  ( q  |`  a ) }   &    |-  P  =  |^| D   &    |-  C  =  { f  |  E. g  e.  L  E. h  e.  R  E. a  e.  P  ( ( g  |`  a )  =  f  /\  ( h  |`  a )  =  f ) }   &    |-  M  =  U. C   =>    |-  ( ( ( ( L  C_  No  /\  R  C_ 
 No )  /\  ( L  e.  A  /\  R  e.  B )
 )  /\  A. x  e.  L  A. y  e.  R  x < s
 y )  ->  M  e.  No )
 
Theoremaxfelem22 23535* Lemma for axfe (future) . Given an element  K of  P, there are elements  l and  r of  L and  R whose restrictions to  K are a prefix of  M. (Contributed by Scott Fenton, 23-Apr-2012.)
 |-  D  =  { a  e.  On  |  A. p  e.  L  A. q  e.  R  ( p  |`  a )  =/=  ( q  |`  a ) }   &    |-  P  =  |^| D   &    |-  C  =  { f  |  E. g  e.  L  E. h  e.  R  E. a  e.  P  ( ( g  |`  a )  =  f  /\  ( h  |`  a )  =  f ) }   &    |-  M  =  U. C   =>    |-  ( ( ( ( L  C_  No  /\  R  C_ 
 No )  /\  ( L  e.  A  /\  R  e.  B )
 )  /\  A. x  e.  L  A. y  e.  R  x < s
 y )  ->  ( K  e.  P  ->  ( E. l  e.  L  ( l  |`  K ) 
 C_  M  /\  E. r  e.  R  (
 r  |`  K )  C_  M ) ) )
 
16.6.27  Symmetric difference
 
Syntaxcsymdif 23536 Declare the syntax for symmetric difference.
 class  ( A(++)
 B )
 
Definitiondf-symdif 23537 Define the symmetric difference of two classes. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  ( A(++) B )  =  ( ( A  \  B )  u.  ( B  \  A ) )
 
Theoremsymdifcom 23538 Symmetric difference commutes. (Contributed by Scott Fenton, 24-Apr-2012.)
 |-  ( A(++) B )  =  ( B(++) A )
 
Theoremsymdifeq1 23539 Equality theorem for symmetric difference. (Contributed by Scott Fenton, 24-Apr-2012.)
 |-  ( A  =  B  ->  ( A(++) C )  =  ( B(++) C ) )
 
Theoremsymdifeq2 23540 Equality theorem for symmetric difference. (Contributed by Scott Fenton, 24-Apr-2012.)
 |-  ( A  =  B  ->  ( C(++) A )  =  ( C(++) B ) )
 
Theoremnfsymdif 23541 Hypothesis builder for symmetric difference. (Contributed by Scott Fenton, 19-Feb-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A(++) B )
 
Theoremelsymdif 23542 Membership in a symmetric difference. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  ( A  e.  ( B(++) C )  <->  -.  ( A  e.  B 
 <->  A  e.  C ) )
 
Theoremsymdif0 23543 Symmetric difference with the empty class. (Contributed by Scott Fenton, 24-Apr-2012.)
 |-  ( A(++) (/) )  =  A
 
TheoremsymdifV 23544 Symmetric difference with the universal class. (Contributed by Scott Fenton, 24-Apr-2012.)
 |-  ( A(++) _V )  =  ( _V  \  A )
 
Theoremsymdifid 23545 Symmetric difference yields the empty class with the same argument twice. (Contributed by Scott Fenton, 25-Apr-2012.)
 |-  ( A(++) A )  =  (/)
 
Theoremsymdifass 23546 Symmetric difference associates. (Contributed by Scott Fenton, 24-Apr-2012.)
 |-  ( A(++) ( B(++) C ) )  =  ( ( A(++) B )(++) C )
 
Theorembrsymdif 23547 The binary relationship of a symmetric difference. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  ( A ( R(++) S ) B  <->  -.  ( A R B 
 <->  A S B ) )
 
16.6.28  Quantifier-free definitions
 
Syntaxctxp 23548 Declare the syntax for tail cross product.
 class  ( A 
 (x)  B )
 
Syntaxcpprod 23549 Declare the syntax for the parallel product.
 class pprod ( R ,  S )
 
Syntaxcsset 23550 Declare the subset relationship class.
 class  SSet
 
Syntaxctrans 23551 Declare the transitive set class.
 class  Trans
 
Syntaxcbigcup 23552 Declare the set union relationship.
 class  Bigcup
 
Syntaxcfix 23553 Declare the syntax for the fixpoints of a class.
 class  Fix A
 
Syntaxclimits 23554 Declare the class of limit ordinals.
 class  Limits
 
Syntaxcfuns 23555 Declare the syntax for the class of all function.
 class  Funs
 
Syntaxcsingle 23556 Declare the syntax for the singleton function.
 class Singleton
 
Syntaxcsingles 23557 Declare the syntax for the class of all singletons.
 class  Singletons
 
Syntaxcimage 23558 Declare the syntax for the image functor.
 class Image A
 
Syntaxccart 23559 Declare the syntax for the cartesian function.
 class Cart
 
Syntaxcimg 23560 Declare the syntax for the image function.
 class Img
 
Syntaxcdomain 23561 Declare the syntax for the domain function.
 class Domain
 
Syntaxcrange 23562 Declare the syntax for the range function.
 class Range
 
Syntaxcapply 23563 Declare the syntax for the application function.
 class Apply
 
Syntaxccup 23564 Declare the syntax for the cup function.
 class Cup
 
Syntaxccap 23565 Declare the syntax for the cap function.
 class Cap
 
Syntaxcsuccf 23566 Declare the syntax for the successor function.
 class Succ
 
Syntaxcfunpart 23567 Declare the syntax for the functional part functor.
 class Funpart F
 
Syntaxcfullfn 23568 Declare the syntax for the full function functor.
 class FullFun F
 
Syntaxcrestrict 23569 Declare the syntax for the restriction function.
 class Restrict
 
Definitiondf-txp 23570 Define the tail cross of two classes. Membership in this class is defined by txpss3v 23593 and brtxp 23595. (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 23571 Define the parallel product of two classes. Membership in this class is defined by pprodss4v 23599 and brpprod 23600. (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 23572 Define the subset class. For the value, see brsset 23604. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  SSet  =  ( ( _V  X.  _V )  \  ran  (  _E  (x)  ( _V  \  _E  ) ) )
 
Definitiondf-trans 23573 Define the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  Trans  =  ( _V  \  ran  (
 (  _E  o.  _E  )  \  _E  ) )
 
Definitiondf-bigcup 23574 Define the Bigcup function, which, per fvbigcup 23617, 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 23575 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 23576 Define the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  Limits  =  ( ( On  i^i  Fix Bigcup )  \  { (/) } )
 
Definitiondf-funs 23577 Define the class of all functions. See elfuns 23628 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 23578 Define the singleton function. See brsingle 23630 for its value. (Contributed by Scott Fenton, 4-Apr-2014.)
 |- Singleton  =  ( ( _V  X.  _V )  \  ran  ( ( _V  (x)  _E  )(++) (  _I  (x)  _V )
 ) )
 
Definitiondf-singles 23579 Define the class of all singletons. See elsingles 23631 for membership. (Contributed by Scott Fenton, 19-Feb-2013.)
 |-  Singletons  =  ran Singleton
 
Definitiondf-image 23580 Define the image functor. This function takes a set  A to a function  x  |->  ( A
" x ), providing that the latter exists. See imageval 23643 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 23581 Define the cartesian product function. See brcart 23645 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 23582 Define the image function. See brimg 23650 for its value. (Contributed by Scott Fenton, 12-Apr-2014.)
 |- Img  =  (Image ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) )  o. Cart
 )
 
Definitiondf-domain 23583 Define the domain function. See brdomain 23646 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
 |- Domain  = Image ( 1st  |`  ( _V  X.  _V ) )
 
Definitiondf-range 23584 Define the range function. See brrange 23647 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
 |- Range  = Image ( 2nd  |`  ( _V  X.  _V ) )
 
Definitiondf-cup 23585 Define the little cup function. See brcup 23652 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 23586 Define the little cap function. See brcap 23653 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 23587 Define the restriction function. See brrestrict 23661 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
 |- Restrict  =  (Cap 
 o.  ( 1st  (x)  (Cart  o.  ( 2nd  (x)  (Range  o. 
 1st ) ) ) ) )
 
Definitiondf-succf 23588 Define the successor function. See brsuccf 23654 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
 |- Succ  =  (Cup 
 o.  (  _I  (x) Singleton ) )
 
Definitiondf-apply 23589 Define the application function. See brapply 23651 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 23590 Define the functional part of a class  F. This is the maximal part of  F that is a function. See funpartfun 23655 and funpartfv 23657 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 23591 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 23592 The binary relationship over  _V always holds. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  A _V B
 
Theoremtxpss3v 23593 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 23594 A tail cross product is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  Rel  ( A  (x)  B )
 
Theorembrtxp 23595 Characterize a trinary relationship over a tail cross product. Together with txpss3v 23593, 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 23596* 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 23597 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 23598 A converse law for parallel product. (Contributed by Scott Fenton, 3-May-2014.)
 |- pprod ( R ,  S )  =  `'pprod ( `' R ,  `' S )
 
Theorempprodss4v 23599 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 23600 Characterize a quatary relationship over a tail cross product. Together with pprodss4v 23599, 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 ) )
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