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Theorem List for Metamath Proof Explorer - 29101-29200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiprodgam 29101* An infinite product version of Euler's gamma function. (Contributed by Scott Fenton, 12-Feb-2018.)
 |-  ( ph  ->  A  e.  ( CC  \  ( ZZ  \  NN ) ) )   =>    |-  ( ph  ->  (
 _G `  A )  =  ( prod_ k  e.  NN  ( ( ( 1  +  ( 1  /  k ) )  ^c  A )  /  (
 1  +  ( A 
 /  k ) ) )  /  A ) )
 
21.8.7  Falling and Rising Factorial
 
Syntaxcfallfac 29102 Declare the syntax for the falling factorial.
 class FallFac
 
Syntaxcrisefac 29103 Declare the syntax for the rising factorial.
 class RiseFac
 
Definitiondf-risefac 29104* Define the rising factorial function. This is the function  ( A  x.  ( A  +  1
)  x.  ... ( A  +  N )
) for complex  A and nonnegative integers  N. (Contributed by Scott Fenton, 5-Jan-2018.)
 |- RiseFac  =  ( x  e.  CC ,  n  e.  NN0  |->  prod_ k  e.  ( 0 ... ( n  -  1 ) ) ( x  +  k
 ) )
 
Definitiondf-fallfac 29105* Define the falling factorial function. This is the function  ( A  x.  ( A  -  1
)  x.  ... ( A  -  N )
) for complex  A and nonnegative integers  N. (Contributed by Scott Fenton, 5-Jan-2018.)
 |- FallFac  =  ( x  e.  CC ,  n  e.  NN0  |->  prod_ k  e.  ( 0 ... ( n  -  1 ) ) ( x  -  k
 ) )
 
Theoremrisefacval 29106* The value of the rising factorial function. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  =  prod_ k  e.  (
 0 ... ( N  -  1 ) ) ( A  +  k ) )
 
Theoremfallfacval 29107* The value of the falling factorial function. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A FallFac  N )  =  prod_ k  e.  (
 0 ... ( N  -  1 ) ) ( A  -  k ) )
 
Theoremrisefacval2 29108* One-based value of rising factorial. (Contributed by Scott Fenton, 15-Jan-2018.)
 |-  (
 ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  =  prod_ k  e.  (
 1 ... N ) ( A  +  ( k  -  1 ) ) )
 
Theoremfallfacval2 29109* One-based value of falling factorial. (Contributed by Scott Fenton, 15-Jan-2018.)
 |-  (
 ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A FallFac  N )  =  prod_ k  e.  (
 1 ... N ) ( A  -  ( k  -  1 ) ) )
 
Theoremfallfacval3 29110* A product representation of falling factorial when  A is a nonnegative integer. (Contributed by Scott Fenton, 20-Mar-2018.)
 |-  ( N  e.  ( 0 ... A )  ->  ( A FallFac  N )  =  prod_ k  e.  ( ( A  -  ( N  -  1 ) ) ... A ) k )
 
Theoremrisefaccllem 29111* Lemma for rising factorial closure laws. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  S  C_ 
 CC   &    |-  1  e.  S   &    |-  (
 ( x  e.  S  /\  y  e.  S )  ->  ( x  x.  y )  e.  S )   &    |-  ( ( A  e.  S  /\  k  e.  NN0 )  ->  ( A  +  k )  e.  S )   =>    |-  ( ( A  e.  S  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  e.  S )
 
Theoremfallfaccllem 29112* Lemma for falling factorial closure laws. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  S  C_ 
 CC   &    |-  1  e.  S   &    |-  (
 ( x  e.  S  /\  y  e.  S )  ->  ( x  x.  y )  e.  S )   &    |-  ( ( A  e.  S  /\  k  e.  NN0 )  ->  ( A  -  k )  e.  S )   =>    |-  ( ( A  e.  S  /\  N  e.  NN0 )  ->  ( A FallFac  N )  e.  S )
 
Theoremrisefaccl 29113 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  e.  CC )
 
Theoremfallfaccl 29114 Closure law for falling factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A FallFac  N )  e.  CC )
 
Theoremrerisefaccl 29115 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 ( A  e.  RR  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  e.  RR )
 
Theoremrefallfaccl 29116 Closure law for falling factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 ( A  e.  RR  /\  N  e.  NN0 )  ->  ( A FallFac  N )  e.  RR )
 
Theoremnnrisefaccl 29117 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 ( A  e.  NN  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  e.  NN )
 
Theoremzrisefaccl 29118 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 ( A  e.  ZZ  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  e.  ZZ )
 
Theoremzfallfaccl 29119 Closure law for falling factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 ( A  e.  ZZ  /\  N  e.  NN0 )  ->  ( A FallFac  N )  e.  ZZ )
 
Theoremnn0risefaccl 29120 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 ( A  e.  NN0  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  e.  NN0 )
 
Theoremrprisefaccl 29121 Closure law for rising factorial. (Contributed by Scott Fenton, 9-Jan-2018.)
 |-  (
 ( A  e.  RR+  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  e.  RR+ )
 
Theoremrisefallfac 29122 A relationship between rising and falling factorials. (Contributed by Scott Fenton, 15-Jan-2018.)
 |-  (
 ( X  e.  CC  /\  N  e.  NN0 )  ->  ( X RiseFac  N )  =  ( ( -u 1 ^ N )  x.  ( -u X FallFac  N ) ) )
 
Theoremfallrisefac 29123 A relationship between falling and rising factorials. (Contributed by Scott Fenton, 17-Jan-2018.)
 |-  (
 ( X  e.  CC  /\  N  e.  NN0 )  ->  ( X FallFac  N )  =  ( ( -u 1 ^ N )  x.  ( -u X RiseFac  N ) ) )
 
Theoremrisefall0lem 29124 Lemma for risefac0 29125 and fallfac0 29126. Show a particular set of finite integers is empty. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 0 ... ( 0  -  1 ) )  =  (/)
 
Theoremrisefac0 29125 The value of the rising factorial when  N  =  0. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( A  e.  CC  ->  ( A RiseFac  0 )  =  1 )
 
Theoremfallfac0 29126 The value of the falling factorial when  N  =  0. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( A  e.  CC  ->  ( A FallFac  0 )  =  1 )
 
Theoremrisefacp1 29127 The value of the rising factorial at a successor. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A RiseFac  ( N  +  1 ) )  =  ( ( A RiseFac  N )  x.  ( A  +  N )
 ) )
 
Theoremfallfacp1 29128 The value of the falling factorial at a successor. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A FallFac  ( N  +  1 ) )  =  ( ( A FallFac  N )  x.  ( A  -  N ) ) )
 
Theoremrisefacp1d 29129 The value of the rising factorial at a successor. (Contributed by Scott Fenton, 19-Mar-2018.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( A RiseFac  ( N  +  1 ) )  =  ( ( A RiseFac  N )  x.  ( A  +  N ) ) )
 
Theoremfallfacp1d 29130 The value of the falling factorial at a successor. (Contributed by Scott Fenton, 19-Mar-2018.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( A FallFac  ( N  +  1 ) )  =  ( ( A FallFac  N )  x.  ( A  -  N ) ) )
 
Theoremrisefac1 29131 The value of rising factorial at one. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( A  e.  CC  ->  ( A RiseFac  1 )  =  A )
 
Theoremfallfac1 29132 The value of falling factorial at one. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( A  e.  CC  ->  ( A FallFac  1 )  =  A )
 
Theoremrisefacfac 29133 Relate rising factorial to factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( N  e.  NN0  ->  (
 1 RiseFac  N )  =  ( ! `  N ) )
 
Theoremfallfacfwd 29134 The forward difference of a falling factorial. (Contributed by Scott Fenton, 21-Jan-2018.)
 |-  (
 ( A  e.  CC  /\  N  e.  NN )  ->  ( ( ( A  +  1 ) FallFac  N )  -  ( A FallFac  N ) )  =  ( N  x.  ( A FallFac  ( N  -  1 ) ) ) )
 
Theorem0fallfac 29135 The value of the zero falling factorial at natural  N. (Contributed by Scott Fenton, 17-Feb-2018.)
 |-  ( N  e.  NN  ->  ( 0 FallFac  N )  =  0 )
 
Theorem0risefac 29136 The value of the zero rising factorial at natural  N. (Contributed by Scott Fenton, 17-Feb-2018.)
 |-  ( N  e.  NN  ->  ( 0 RiseFac  N )  =  0 )
 
Theorembinomfallfaclem1 29137 Lemma for binomfallfac 29139. Closure law. (Contributed by Scott Fenton, 13-Mar-2018.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ( ph  /\  K  e.  ( 0 ... N ) )  ->  ( ( N  _C  K )  x.  ( ( A FallFac  ( N  -  K ) )  x.  ( B FallFac  ( K  +  1 ) ) ) )  e.  CC )
 
Theorembinomfallfaclem2 29138* Lemma for binomfallfac 29139. Inductive step. (Contributed by Scott Fenton, 13-Mar-2018.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ps  ->  ( ( A  +  B ) FallFac  N )  =  sum_ k  e.  ( 0 ...
 N ) ( ( N  _C  k )  x.  ( ( A FallFac  ( N  -  k
 ) )  x.  ( B FallFac  k ) ) ) )   =>    |-  ( ( ph  /\  ps )  ->  ( ( A  +  B ) FallFac  ( N  +  1 )
 )  =  sum_ k  e.  ( 0 ... ( N  +  1 )
 ) ( ( ( N  +  1 )  _C  k )  x.  ( ( A FallFac  (
 ( N  +  1 )  -  k ) )  x.  ( B FallFac  k ) ) ) )
 
Theorembinomfallfac 29139* A version of the binomial theorem using falling factorials instead of exponentials. (Contributed by Scott Fenton, 13-Mar-2018.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  N  e.  NN0 )  ->  ( ( A  +  B ) FallFac  N )  = 
 sum_ k  e.  (
 0 ... N ) ( ( N  _C  k
 )  x.  ( ( A FallFac  ( N  -  k ) )  x.  ( B FallFac  k )
 ) ) )
 
Theorembinomrisefac 29140* A version of the binomial theorem using rising factorials instead of exponentials. (Contributed by Scott Fenton, 16-Mar-2018.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  N  e.  NN0 )  ->  ( ( A  +  B ) RiseFac  N )  = 
 sum_ k  e.  (
 0 ... N ) ( ( N  _C  k
 )  x.  ( ( A RiseFac  ( N  -  k ) )  x.  ( B RiseFac  k )
 ) ) )
 
Theoremfallfacval4 29141 Represent the falling factorial via factorials when the first argument is a natural. (Contributed by Scott Fenton, 20-Mar-2018.)
 |-  ( N  e.  ( 0 ... A )  ->  ( A FallFac  N )  =  ( ( ! `  A )  /  ( ! `  ( A  -  N ) ) ) )
 
Theorembcfallfac 29142 Binomial coefficient in terms of falling factorials. (Contributed by Scott Fenton, 20-Mar-2018.)
 |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  =  ( ( N FallFac  K ) 
 /  ( ! `  K ) ) )
 
Theoremfallfacfac 29143 Relate falling factorial to factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( N  e.  NN0  ->  ( N FallFac  N )  =  ( ! `  N ) )
 
21.8.8  Factorial limits
 
Theoremfaclimlem1 29144* Lemma for faclim 29147. Closed form for a particular sequence. (Contributed by Scott Fenton, 15-Dec-2017.)
 |-  ( M  e.  NN0  ->  seq 1
 (  x.  ,  ( n  e.  NN  |->  ( ( ( 1  +  ( M  /  n ) )  x.  ( 1  +  ( 1  /  n ) ) )  /  ( 1  +  (
 ( M  +  1 )  /  n ) ) ) ) )  =  ( x  e. 
 NN  |->  ( ( M  +  1 )  x.  ( ( x  +  1 )  /  ( x  +  ( M  +  1 ) ) ) ) ) )
 
Theoremfaclimlem2 29145* Lemma for faclim 29147. Show a limit for the inductive step. (Contributed by Scott Fenton, 15-Dec-2017.)
 |-  ( M  e.  NN0  ->  seq 1
 (  x.  ,  ( n  e.  NN  |->  ( ( ( 1  +  ( M  /  n ) )  x.  ( 1  +  ( 1  /  n ) ) )  /  ( 1  +  (
 ( M  +  1 )  /  n ) ) ) ) )  ~~>  ( M  +  1
 ) )
 
Theoremfaclimlem3 29146 Lemma for faclim 29147. Algebraic manipulation for the final induction. (Contributed by Scott Fenton, 15-Dec-2017.)
 |-  (
 ( M  e.  NN0  /\  B  e.  NN )  ->  ( ( ( 1  +  ( 1  /  B ) ) ^
 ( M  +  1 ) )  /  (
 1  +  ( ( M  +  1 ) 
 /  B ) ) )  =  ( ( ( ( 1  +  ( 1  /  B ) ) ^ M )  /  ( 1  +  ( M  /  B ) ) )  x.  ( ( ( 1  +  ( M  /  B ) )  x.  ( 1  +  (
 1  /  B )
 ) )  /  (
 1  +  ( ( M  +  1 ) 
 /  B ) ) ) ) )
 
Theoremfaclim 29147* An infinite product expression relating to factorials. Originally due to Euler. (Contributed by Scott Fenton, 22-Nov-2017.)
 |-  F  =  ( n  e.  NN  |->  ( ( ( 1  +  ( 1  /  n ) ) ^ A )  /  (
 1  +  ( A 
 /  n ) ) ) )   =>    |-  ( A  e.  NN0  ->  seq 1 (  x.  ,  F )  ~~>  ( ! `  A ) )
 
Theoremiprodfac 29148* An infinite product expression for factorial. (Contributed by Scott Fenton, 15-Dec-2017.)
 |-  ( A  e.  NN0  ->  ( ! `  A )  = 
 prod_ k  e.  NN  ( ( ( 1  +  ( 1  /  k ) ) ^ A )  /  (
 1  +  ( A 
 /  k ) ) ) )
 
Theoremfaclim2 29149* Another factorial limit due to Euler. (Contributed by Scott Fenton, 17-Dec-2017.)
 |-  F  =  ( n  e.  NN  |->  ( ( ( ! `
  n )  x.  ( ( n  +  1 ) ^ M ) )  /  ( ! `  ( n  +  M ) ) ) )   =>    |-  ( M  e.  NN0  ->  F 
 ~~>  1 )
 
21.8.9  Greatest common divisor and divisibility
 
Theorempdivsq 29150 Condition for a prime dividing a square. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( P  e.  Prime  /\  M  e.  ZZ )  ->  ( P  ||  M  <->  P 
 ||  ( M ^
 2 ) ) )
 
Theoremdvdspw 29151 Exponentiation law for divisibility. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  ->  ( K  ||  M  ->  K  ||  ( M ^ N ) ) )
 
Theoremgcd32 29152 Swap the second and third arguments of a gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( A  gcd  B )  gcd  C )  =  ( ( A 
 gcd  C )  gcd  B ) )
 
Theoremgcdabsorb 29153 Absorption law for gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  gcd  B )  =  ( A  gcd  B ) )
 
21.8.10  Properties of relationships
 
Theorembrtp 29154 A condition for a binary relation over an unordered triple. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  X  e.  _V   &    |-  Y  e.  _V   =>    |-  ( X { <. A ,  B >. ,  <. C ,  D >. ,  <. E ,  F >. } Y  <->  ( ( X  =  A  /\  Y  =  B )  \/  ( X  =  C  /\  Y  =  D )  \/  ( X  =  E  /\  Y  =  F ) ) )
 
Theoremdftr6 29155 A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012.)
 |-  A  e.  _V   =>    |-  ( Tr  A  <->  A  e.  ( _V  \  ran  ( (  _E  o.  _E  )  \  _E  ) ) )
 
Theoremcoep 29156* Composition with epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A (  _E  o.  R ) B  <->  E. x  e.  B  A R x )
 
Theoremcoepr 29157* Composition with the converse of epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A ( R  o.  `'  _E  ) B  <->  E. x  e.  A  x R B )
 
Theoremdffr5 29158 A quantifier free definition of a well-founded relationship. (Contributed by Scott Fenton, 11-Apr-2011.)
 |-  ( R  Fr  A  <->  ( ~P A  \  { (/) } )  C_  ran  (  _E  \  (  _E  o.  `' R ) ) )
 
Theoremdfso2 29159 Quantifier free definition of a strict order. (Contributed by Scott Fenton, 22-Feb-2013.)
 |-  ( R  Or  A  <->  ( R  Po  A  /\  ( A  X.  A )  C_  ( R  u.  (  _I  u.  `' R ) ) ) )
 
Theoremdfpo2 29160 Quantifier free definition of a partial ordering. (Contributed by Scott Fenton, 22-Feb-2013.)
 |-  ( R  Po  A  <->  ( ( R  i^i  (  _I  |`  A ) )  =  (/)  /\  (
 ( R  i^i  ( A  X.  A ) )  o.  ( R  i^i  ( A  X.  A ) ) )  C_  R ) )
 
Theorembr8 29161* Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  (
 a  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 b  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 c  =  C  ->  ( ch  <->  th ) )   &    |-  (
 d  =  D  ->  ( th  <->  ta ) )   &    |-  (
 e  =  E  ->  ( ta  <->  et ) )   &    |-  (
 f  =  F  ->  ( et  <->  ze ) )   &    |-  (
 g  =  G  ->  ( ze  <->  si ) )   &    |-  ( h  =  H  ->  (
 si 
 <->  rh ) )   &    |-  ( x  =  X  ->  P  =  Q )   &    |-  R  =  { <. p ,  q >.  |  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  E. e  e.  P  E. f  e.  P  E. g  e.  P  E. h  e.  P  ( p  = 
 <. <. a ,  b >. ,  <. c ,  d >.
 >.  /\  q  =  <. <.
 e ,  f >. , 
 <. g ,  h >. >.  /\  ph ) }   =>    |-  ( ( ( X  e.  S  /\  A  e.  Q  /\  B  e.  Q )  /\  ( C  e.  Q  /\  D  e.  Q  /\  E  e.  Q )  /\  ( F  e.  Q  /\  G  e.  Q  /\  H  e.  Q )
 )  ->  ( <. <. A ,  B >. , 
 <. C ,  D >. >. R <. <. E ,  F >. ,  <. G ,  H >.
 >. 
 <->  rh ) )
 
Theorembr6 29162* Substitution for a six-place predicate. (Contributed by Scott Fenton, 4-Oct-2013.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  (
 a  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 b  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 c  =  C  ->  ( ch  <->  th ) )   &    |-  (
 d  =  D  ->  ( th  <->  ta ) )   &    |-  (
 e  =  E  ->  ( ta  <->  et ) )   &    |-  (
 f  =  F  ->  ( et  <->  ze ) )   &    |-  ( x  =  X  ->  P  =  Q )   &    |-  R  =  { <. p ,  q >.  |  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  E. e  e.  P  E. f  e.  P  ( p  = 
 <. a ,  <. b ,  c >. >.  /\  q  =  <. d ,  <. e ,  f >. >.  /\  ph ) }   =>    |-  (
 ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q  /\  C  e.  Q )  /\  ( D  e.  Q  /\  E  e.  Q  /\  F  e.  Q )
 )  ->  ( <. A ,  <. B ,  C >.
 >. R <. D ,  <. E ,  F >. >.  <->  ze ) )
 
Theorembr4 29163* Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.)
 |-  (
 a  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 b  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 c  =  C  ->  ( ch  <->  th ) )   &    |-  (
 d  =  D  ->  ( th  <->  ta ) )   &    |-  ( x  =  X  ->  P  =  Q )   &    |-  R  =  { <. p ,  q >.  |  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  ( p  = 
 <. a ,  b >.  /\  q  =  <. c ,  d >.  /\  ph ) }   =>    |-  ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q )  /\  ( C  e.  Q  /\  D  e.  Q ) )  ->  ( <. A ,  B >. R <. C ,  D >.  <->  ta ) )
 
Theoremdfres3 29164 Alternate definition of restriction. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  ran  A )
 )
 
Theoremcnvco1 29165 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
 |-  `' ( `' A  o.  B )  =  ( `' B  o.  A )
 
Theoremcnvco2 29166 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
 |-  `' ( A  o.  `' B )  =  ( B  o.  `' A )
 
Theoremeldm3 29167 Quantifier-free definition of membership in a domain. (Contributed by Scott Fenton, 21-Jan-2017.)
 |-  ( A  e.  dom  B  <->  ( B  |`  { A } )  =/=  (/) )
 
Theoremelrn3 29168 Quantifier-free definition of membership in a range. (Contributed by Scott Fenton, 21-Jan-2017.)
 |-  ( A  e.  ran  B  <->  ( B  i^i  ( _V  X.  { A } ) )  =/=  (/) )
 
Theorempocnv 29169 The converse of a partial ordering is still a partial ordering. (Contributed by Scott Fenton, 13-Jun-2018.)
 |-  ( R  Po  A  ->  `' R  Po  A )
 
Theoremsocnv 29170 The converse of a strict ordering is still a strict ordering. (Contributed by Scott Fenton, 13-Jun-2018.)
 |-  ( R  Or  A  ->  `' R  Or  A )
 
21.8.11  Properties of functions and mappings
 
Theoremfunpsstri 29171 A condition for subset trichotomy for functions. (Contributed by Scott Fenton, 19-Apr-2011.)
 |-  (
 ( Fun  H  /\  ( F  C_  H  /\  G  C_  H )  /\  ( dom  F  C_  dom  G  \/  dom  G  C_  dom  F ) )  ->  ( F  C.  G  \/  F  =  G  \/  G  C.  F ) )
 
Theoremfundmpss 29172 If a class  F is a proper subset of a function  G, then  dom  F  C.  dom  G. (Contributed by Scott Fenton, 20-Apr-2011.)
 |-  ( Fun  G  ->  ( F  C.  G  ->  dom  F  C.  dom 
 G ) )
 
Theoremfvresval 29173 The value of a function at a restriction is either null or the same as the function itself. (Contributed by Scott Fenton, 4-Sep-2011.)
 |-  (
 ( ( F  |`  B ) `
  A )  =  ( F `  A )  \/  ( ( F  |`  B ) `  A )  =  (/) )
 
Theoremmptrel 29174 The maps-to notation always describes a relationship. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Rel  ( x  e.  A  |->  B )
 
Theoremfunsseq 29175 Given two functions with equal domains, equality only requires one direction of the subset relationship. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof shortened by Mario Carneiro, 3-May-2015.)
 |-  (
 ( Fun  F  /\  Fun 
 G  /\  dom  F  =  dom  G )  ->  ( F  =  G  <->  F  C_  G ) )
 
Theoremfununiq 29176 The uniqueness condition of functions. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( Fun  F  ->  ( ( A F B  /\  A F C ) 
 ->  B  =  C ) )
 
Theoremfunbreq 29177 An equality condition for functions. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( ( Fun  F  /\  A F B ) 
 ->  ( A F C  <->  B  =  C ) )
 
Theoremmpteq12d 29178 An equality inference for the maps to notation. Compare mpteq12dv 4515. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/ x ph   &    |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )
 
Theoremfprb 29179* A condition for functionhood over a pair. (Contributed by Scott Fenton, 16-Sep-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  =/=  B  ->  ( F : { A ,  B } --> R  <->  E. x  e.  R  E. y  e.  R  F  =  { <. A ,  x >. ,  <. B ,  y >. } ) )
 
Theorembr1steq 29180 Uniqueness condition for binary relationship over the  1st relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >. 1st C  <->  C  =  A )
 
Theorembr2ndeq 29181 Uniqueness condition for binary relationship over the  2nd relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >. 2nd C  <->  C  =  B )
 
Theoremdfdm5 29182 Definition of domain in terms of 
1st and image. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  dom  A  =  ( ( 1st  |`  ( _V  X.  _V ) ) " A )
 
Theoremdfrn5 29183 Definition of range in terms of 
2nd and image. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ran  A  =  ( ( 2nd  |`  ( _V  X.  _V ) ) " A )
 
Theoremopelco3 29184 Alternate way of saying that an ordered pair is in a composition. (Contributed by Scott Fenton, 6-May-2018.)
 |-  ( <. A ,  B >.  e.  ( C  o.  D ) 
 <->  B  e.  ( C
 " ( D " { A } ) ) )
 
Theoremelima4 29185 Quantifier-free expression saying that a class is a member of an image. (Contributed by Scott Fenton, 8-May-2018.)
 |-  ( A  e.  ( R " B )  <->  ( R  i^i  ( B  X.  { A } ) )  =/=  (/) )
 
21.8.12  Epsilon induction
 
Theoremsetinds 29186* Principle of  _E induction (set induction). If a property passes from all elements of  x to  x itself, then it holds for all  x. (Contributed by Scott Fenton, 10-Mar-2011.)
 |-  ( A. y  e.  x  [. y  /  x ]. ph 
 ->  ph )   =>    |-  ph
 
Theoremsetinds2f 29187*  _E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/ x ps   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   &    |-  ( A. y  e.  x  ps  ->  ph )   =>    |-  ph
 
Theoremsetinds2 29188*  _E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ( A. y  e.  x  ps  ->  ph )   =>    |-  ph
 
21.8.13  Ordinal numbers
 
Theoremelpotr 29189* A class of transitive sets is partially ordered by  _E. (Contributed by Scott Fenton, 15-Oct-2010.)
 |-  ( A. z  e.  A  Tr  z  ->  _E  Po  A )
 
Theoremdford5reg 29190 Given ax-reg 8021, an ordinal is a transitive class totally ordered by epsilon. (Contributed by Scott Fenton, 28-Jan-2011.)
 |-  ( Ord  A  <->  ( Tr  A  /\  _E  Or  A ) )
 
Theoremdfon2lem1 29191 Lemma for dfon2 29200. (Contributed by Scott Fenton, 28-Feb-2011.)
 |-  Tr  U.
 { x  |  (
 ph  /\  Tr  x  /\  ps ) }
 
Theoremdfon2lem2 29192* Lemma for dfon2 29200 (Contributed by Scott Fenton, 28-Feb-2011.)
 |-  U. { x  |  ( x  C_  A  /\  ph  /\  ps ) }  C_  A
 
Theoremdfon2lem3 29193* Lemma for dfon2 29200. All sets satisfying the new definition are transitive and untangled. (Contributed by Scott Fenton, 25-Feb-2011.)
 |-  ( A  e.  V  ->  (
 A. x ( ( x  C.  A  /\  Tr  x )  ->  x  e.  A )  ->  ( Tr  A  /\  A. z  e.  A  -.  z  e.  z ) ) )
 
Theoremdfon2lem4 29194* Lemma for dfon2 29200. If two sets satisfy the new definition, then one is a subset of the other. (Contributed by Scott Fenton, 25-Feb-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  (
 ( A. x ( ( x  C.  A  /\  Tr  x )  ->  x  e.  A )  /\  A. y ( ( y  C.  B  /\  Tr  y
 )  ->  y  e.  B ) )  ->  ( A  C_  B  \/  B  C_  A ) )
 
Theoremdfon2lem5 29195* Lemma for dfon2 29200. Two sets satisfying the new definition also satisfy trichotomy with respect to 
e.. (Contributed by Scott Fenton, 25-Feb-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  (
 ( A. x ( ( x  C.  A  /\  Tr  x )  ->  x  e.  A )  /\  A. y ( ( y  C.  B  /\  Tr  y
 )  ->  y  e.  B ) )  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A )
 )
 
Theoremdfon2lem6 29196* Lemma for dfon2 29200. A transitive class of sets satisfying the new definition satisfies the new definition. (Contributed by Scott Fenton, 25-Feb-2011.)
 |-  (
 ( Tr  S  /\  A. x  e.  S  A. z ( ( z  C.  x  /\  Tr  z
 )  ->  z  e.  x ) )  ->  A. y ( ( y  C.  S  /\  Tr  y
 )  ->  y  e.  S ) )
 
Theoremdfon2lem7 29197* Lemma for dfon2 29200. All elements of a new ordinal are new ordinals. (Contributed by Scott Fenton, 25-Feb-2011.)
 |-  A  e.  _V   =>    |-  ( A. x ( ( x  C.  A  /\  Tr  x )  ->  x  e.  A )  ->  ( B  e.  A  ->  A. y ( ( y  C.  B  /\  Tr  y
 )  ->  y  e.  B ) ) )
 
Theoremdfon2lem8 29198* Lemma for dfon2 29200. The intersection of a nonempty class  A of new ordinals is itself a new ordinal and is contained within  A (Contributed by Scott Fenton, 26-Feb-2011.)
 |-  (
 ( A  =/=  (/)  /\  A. x  e.  A  A. y
 ( ( y  C.  x  /\  Tr  y ) 
 ->  y  e.  x ) )  ->  ( A. z ( ( z  C.  |^| A  /\  Tr  z )  ->  z  e. 
 |^| A )  /\  |^|
 A  e.  A ) )
 
Theoremdfon2lem9 29199* Lemma for dfon2 29200. A class of new ordinals is well-founded by  _E. (Contributed by Scott Fenton, 3-Mar-2011.)
 |-  ( A. x  e.  A  A. y ( ( y  C.  x  /\  Tr  y
 )  ->  y  e.  x )  ->  _E  Fr  A )
 
Theoremdfon2 29200*  On consists of all sets that contain all its transitive proper subsets. This definition comes from J. R. Isbell, "A Definition of Ordinal Numbers," American Mathematical Monthly, vol 67 (1960), pp. 51-52. (Contributed by Scott Fenton, 20-Feb-2011.)
 |-  On  =  { x  |  A. y ( ( y  C.  x  /\  Tr  y
 )  ->  y  e.  x ) }
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