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Theorem List for Metamath Proof Explorer - 12801-12900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremswrd2lsw 12801 Extract the last two single symbol from a word. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
 |-  ( ( W  e. Word  V 
 /\  1  <  ( # `
  W ) ) 
 ->  ( W substr  <. ( ( # `  W )  -  2 ) ,  ( # `
  W ) >. )  =  <" ( W `
  ( ( # `  W )  -  2
 ) ) ( lastS  `  W ) "> )
 
Theorem2swrd2eqwrdeq 12802 Two words of length at least 2 are equal if and only if they have the same prefix and the same two single symbols suffix. (Contributed by AV, 24-Sep-2018.) (Revised by Mario Carneiro/AV, 23-Oct-2018.)
 |-  ( ( W  e. Word  V 
 /\  U  e. Word  V  /\  1  <  ( # `  W ) )  ->  ( W  =  U  <->  ( ( # `  W )  =  ( # `  U )  /\  ( ( W substr  <. 0 ,  ( ( # `  W )  -  2 ) >. )  =  ( U substr  <. 0 ,  ( ( # `  W )  -  2 ) >. ) 
 /\  ( W `  ( ( # `  W )  -  2 ) )  =  ( U `  ( ( # `  W )  -  2 ) ) 
 /\  ( lastS  `  W )  =  ( lastS  `  U ) ) ) ) )
 
Theoremccatw2s1ccatws2 12803 The concatenation of a word with two singleton words equals the concatenation of the word with the doubleton consisting of the symbols of the two singletons. (Contributed by Mario Carneiro/AV, 21-Oct-2018.)
 |-  ( ( W  e. Word  V 
 /\  X  e.  V  /\  Y  e.  V ) 
 ->  ( ( W ++  <" X "> ) ++  <" Y "> )  =  ( W ++  <" X Y "> ) )
 
Theoremccat2s1fvwALT 12804 Alternate proof of ccat2s1fvw 12551 using words of length 2, see df-s2 12724. A symbol of the concatenation of a word with two single symbols corresponding to the symbol of the word. (Contributed by AV, 22-Sep-2018.) (Proof shortened by Mario Carneiro/AV, 21-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ( ( W  e. Word  V  /\  I  e. 
 NN0  /\  I  <  ( # `
  W ) ) 
 /\  ( X  e.  V  /\  Y  e.  V ) )  ->  ( ( ( W ++  <" X "> ) ++  <" Y "> ) `  I
 )  =  ( W `
  I ) )
 
Theoremwwlktovf 12805* Lemma 1 for wrd2f1tovbij 12809. (Contributed by Alexander van der Vekens, 27-Jul-2018.)
 |-  D  =  { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w `
  0 )  =  P  /\  { ( w `  0 ) ,  ( w `  1
 ) }  e.  X ) }   &    |-  R  =  { n  e.  V  |  { P ,  n }  e.  X }   &    |-  F  =  ( t  e.  D  |->  ( t `  1 ) )   =>    |-  F : D --> R
 
Theoremwwlktovf1 12806* Lemma 2 for wrd2f1tovbij 12809. (Contributed by Alexander van der Vekens, 27-Jul-2018.)
 |-  D  =  { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w `
  0 )  =  P  /\  { ( w `  0 ) ,  ( w `  1
 ) }  e.  X ) }   &    |-  R  =  { n  e.  V  |  { P ,  n }  e.  X }   &    |-  F  =  ( t  e.  D  |->  ( t `  1 ) )   =>    |-  F : D -1-1-> R
 
Theoremwwlktovfo 12807* Lemma 3 for wrd2f1tovbij 12809. (Contributed by Alexander van der Vekens, 27-Jul-2018.)
 |-  D  =  { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w `
  0 )  =  P  /\  { ( w `  0 ) ,  ( w `  1
 ) }  e.  X ) }   &    |-  R  =  { n  e.  V  |  { P ,  n }  e.  X }   &    |-  F  =  ( t  e.  D  |->  ( t `  1 ) )   =>    |-  ( P  e.  V  ->  F : D -onto-> R )
 
Theoremwwlktovf1o 12808* Lemma 4 for wrd2f1tovbij 12809. (Contributed by Alexander van der Vekens, 28-Jul-2018.)
 |-  D  =  { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w `
  0 )  =  P  /\  { ( w `  0 ) ,  ( w `  1
 ) }  e.  X ) }   &    |-  R  =  { n  e.  V  |  { P ,  n }  e.  X }   &    |-  F  =  ( t  e.  D  |->  ( t `  1 ) )   =>    |-  ( P  e.  V  ->  F : D -1-1-onto-> R )
 
Theoremwrd2f1tovbij 12809* There is a bijection between words of length two with a fixed first symbol contained in a pair and the symbols contained in a pair together with the fixed symbol. (Contributed by Alexander van der Vekens, 28-Jul-2018.)
 |-  ( ( V  e.  Y  /\  P  e.  V )  ->  E. f  f : { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w `  0 )  =  P  /\  {
 ( w `  0
 ) ,  ( w `
  1 ) }  e.  X ) } -1-1-onto-> { n  e.  V  |  { P ,  n }  e.  X }
 )
 
5.8  Reflexive and transitive closures of relations

A relation,  R, has the reflexive property if  A R A holds whenever  A is an element which could be related by the the relation, namely elements of its domain and range. Eliminating dummy variables we see that a segment of the identity relation must be a subset of the relation or  (  _I  |`  ( ran  R  u.  dom  R ) )  C_  R. See issref 5293.

A relation,  R, has the transitive property if  A R C holds whenever there exists an intermediate value  B such that both 
A R B and  B R C hold. This can be expressed without dummy variables as  ( R  o.  R )  C_  R. See cotr 5292.

The transitive closure of a relation,  ( t+ `  R ), is the smallest superset of the relation which has the transitive property. Likewise the reflexive-transitive closure,  ( t* `  R ), is the smallest superset which has both the reflexive and transitive properties.

Not to be confused with the transitive closure of a set, trcl 8072, which is a closure relative to a different transitive property, df-tr 4461.

 
5.8.1  The reflexive and transitive properties of relations
 
Theoremcoss12d 12810 Subset deduction for composition of two classes. (Contributed by RP, 24-Dec-2019.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  C 
 C_  D )   =>    |-  ( ph  ->  ( A  o.  C ) 
 C_  ( B  o.  D ) )
 
Theoremtrrelssd 12811 The composition of subclasses of a transitive relation is a subclass of that relation. (Contributed by RP, 24-Dec-2019.)
 |-  ( ph  ->  ( R  o.  R )  C_  R )   &    |-  ( ph  ->  S 
 C_  R )   &    |-  ( ph  ->  T  C_  R )   =>    |-  ( ph  ->  ( S  o.  T )  C_  R )
 
Theoremxpcogend 12812 The most interesting case of the composition of two cross products. (Contributed by RP, 24-Dec-2019.)
 |-  ( ph  ->  ( B  i^i  C )  =/=  (/) )   =>    |-  ( ph  ->  (
 ( C  X.  D )  o.  ( A  X.  B ) )  =  ( A  X.  D ) )
 
Theoremxpcoidgend 12813 If two classes are not disjoint, then the composition of their cross-product with itself is idempotent. (Contributed by RP, 24-Dec-2019.)
 |-  ( ph  ->  ( A  i^i  B )  =/=  (/) )   =>    |-  ( ph  ->  (
 ( A  X.  B )  o.  ( A  X.  B ) )  =  ( A  X.  B ) )
 
Theoremcotr2g 12814* Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr2 12815 for the main application. (Contributed by RP, 22-Mar-2020.)
 |- 
 dom  B  C_  D   &    |-  ( ran  B  i^i  dom  A )  C_  E   &    |-  ran  A  C_  F   =>    |-  (
 ( A  o.  B )  C_  C  <->  A. x  e.  D  A. y  e.  E  A. z  e.  F  (
 ( x B y 
 /\  y A z )  ->  x C z ) )
 
Theoremcotr2 12815* Two ways of saying a relation is transitive. Special instance of cotr2g 12814. (Contributed by RP, 22-Mar-2020.)
 |- 
 dom  R  C_  A   &    |-  ( dom  R  i^i  ran  R )  C_  B   &    |-  ran  R  C_  C   =>    |-  (
 ( R  o.  R )  C_  R  <->  A. x  e.  A  A. y  e.  B  A. z  e.  C  (
 ( x R y 
 /\  y R z )  ->  x R z ) )
 
Theoremcotr3 12816* Two ways of saying a relation is transitive. (Contributed by RP, 22-Mar-2020.)
 |-  A  =  dom  R   &    |-  B  =  ( A  i^i  C )   &    |-  C  =  ran  R   =>    |-  (
 ( R  o.  R )  C_  R  <->  A. x  e.  A  A. y  e.  B  A. z  e.  C  (
 ( x R y 
 /\  y R z )  ->  x R z ) )
 
Theoremcoemptyd 12817 Deduction about composition of classes with no relational content in common. (Contributed by RP, 24-Dec-2019.)
 |-  ( ph  ->  ( dom  A  i^i  ran  B )  =  (/) )   =>    |-  ( ph  ->  ( A  o.  B )  =  (/) )
 
Theoremxptrrel 12818 The cross product is always a transitive relation. (Contributed by RP, 24-Dec-2019.)
 |-  ( ( A  X.  B )  o.  ( A  X.  B ) ) 
 C_  ( A  X.  B )
 
Theorem0trrel 12819 The empty class is a transitive relation. (Contributed by RP, 24-Dec-2019.)
 |-  ( (/)  o.  (/) )  C_  (/)
 
5.8.2  Basic properties of closures
 
Theoremcleq1lem 12820 Equality implies bijection. (Contributed by RP, 9-May-2020.)
 |-  ( A  =  B  ->  ( ( A  C_  C  /\  ph )  <->  ( B  C_  C  /\  ph ) ) )
 
Theoremcleq1 12821* Equality of relations implies equality of closures. (Contributed by RP, 9-May-2020.)
 |-  ( R  =  S  -> 
 |^| { r  |  ( R  C_  r  /\  ph ) }  =  |^| { r  |  ( S 
 C_  r  /\  ph ) } )
 
Theoremclsslem 12822* The closure of a subclass is a subclass of the closure. (Contributed by RP, 16-May-2020.)
 |-  ( R  C_  S  -> 
 |^| { r  |  ( R  C_  r  /\  ph ) }  C_  |^| { r  |  ( S  C_  r  /\  ph ) } )
 
5.8.3  Definitions and basic properties of transitive closures
 
Syntaxctcl 12823 Extend class notation to include the transitive closure symbol.
 class 
 t+
 
Syntaxcrtcl 12824 Extend class notation with reflexive-transitive closure.
 class 
 t*
 
Definitiondf-trcl 12825* Transitive closure of a relation. This is the smallest superset which has the transitive property. (Contributed by FL, 27-Jun-2011.)
 |-  t+  =  ( x  e.  _V  |->  |^| { z  |  ( x 
 C_  z  /\  (
 z  o.  z ) 
 C_  z ) }
 )
 
Definitiondf-rtrcl 12826* Reflexive-transitive closure of a relation. This is the smallest superset which is reflexive property over all elements of its domain and range and has the transitive property. (Contributed by FL, 27-Jun-2011.)
 |-  t*  =  ( x  e.  _V  |->  |^| { z  |  ( (  _I  |`  ( dom  x  u.  ran  x )
 )  C_  z  /\  x  C_  z  /\  (
 z  o.  z ) 
 C_  z ) }
 )
 
Theoremtrcleq1 12827* Equality of relations implies equality of transitive closures. (Contributed by RP, 9-May-2020.)
 |-  ( R  =  S  -> 
 |^| { r  |  ( R  C_  r  /\  ( r  o.  r
 )  C_  r ) }  =  |^| { r  |  ( S  C_  r  /\  ( r  o.  r
 )  C_  r ) } )
 
Theoremtrclsslem 12828* The transitive closure (as a relation) of a subclass is a subclass of the transitive closure. (Contributed by RP, 3-May-2020.)
 |-  ( R  C_  S  -> 
 |^| { r  |  ( R  C_  r  /\  ( r  o.  r
 )  C_  r ) }  C_  |^| { r  |  ( S  C_  r  /\  ( r  o.  r
 )  C_  r ) } )
 
Theoremtrcleq2lem 12829 Equality implies bijection. (Contributed by RP, 5-May-2020.)
 |-  ( A  =  B  ->  ( ( R  C_  A  /\  ( A  o.  A )  C_  A )  <-> 
 ( R  C_  B  /\  ( B  o.  B )  C_  B ) ) )
 
Theoremcvbtrcl 12830* Change of bound variable in class of all transitive relations which are supersets of a relation. (Contributed by RP, 5-May-2020.)
 |- 
 { x  |  ( R  C_  x  /\  ( x  o.  x )  C_  x ) }  =  { y  |  ( R  C_  y  /\  ( y  o.  y
 )  C_  y ) }
 
Theoremtrcleq12lem 12831 Equality implies bijection. (Contributed by RP, 9-May-2020.)
 |-  ( ( R  =  S  /\  A  =  B )  ->  ( ( R 
 C_  A  /\  ( A  o.  A )  C_  A )  <->  ( S  C_  B  /\  ( B  o.  B )  C_  B ) ) )
 
Theoremtrclexlem 12832 Existence of relation implies existence of union with Cartesian product of domain and range. (Contributed by RP, 5-May-2020.)
 |-  ( R  e.  V  ->  ( R  u.  ( dom  R  X.  ran  R ) )  e.  _V )
 
Theoremtrclublem 12833* If a relation exists then the class of transitive relations which are supersets of that relation is not empty. (Contributed by RP, 28-Apr-2020.)
 |-  ( R  e.  V  ->  ( R  u.  ( dom  R  X.  ran  R ) )  e.  { x  |  ( R  C_  x  /\  ( x  o.  x )  C_  x ) }
 )
 
Theoremtrclubi 12834* The Cartesian product of the domain and range of a relation is an upper bound for its transitive closure. (Contributed by RP, 2-Jan-2020.) (Revised by RP, 28-Apr-2020.)
 |- 
 Rel  R   &    |-  R  e.  V   =>    |-  |^| { s  |  ( R  C_  s  /\  ( s  o.  s
 )  C_  s ) }  C_  ( dom  R  X.  ran  R )
 
Theoremtrclubgi 12835* The union with the Cartesian product of its domain and range is an upper bound for a set's transitive closure. (Contributed by RP, 3-Jan-2020.) (Revised by RP, 28-Apr-2020.)
 |-  R  e.  V   =>    |-  |^| { s  |  ( R  C_  s  /\  ( s  o.  s
 )  C_  s ) }  C_  ( R  u.  ( dom  R  X.  ran  R ) )
 
Theoremtrclub 12836* The Cartesian product of the domain and range of a relation is an upper bound for its transitive closure. (Contributed by RP, 17-May-2020.)
 |-  ( ( R  e.  V  /\  Rel  R )  -> 
 |^| { r  |  ( R  C_  r  /\  ( r  o.  r
 )  C_  r ) }  C_  ( dom  R  X.  ran  R ) )
 
Theoremtrclubg 12837* The union with the Cartesian product of its domain and range is an upper bound for a set's transitive closure (as a relation). (Contributed by RP, 17-May-2020.)
 |-  ( R  e.  V  -> 
 |^| { r  |  ( R  C_  r  /\  ( r  o.  r
 )  C_  r ) }  C_  ( R  u.  ( dom  R  X.  ran  R ) ) )
 
Theoremtrclfv 12838* The transitive closure of a relation. (Contributed by RP, 28-Apr-2020.)
 |-  ( R  e.  V  ->  ( t+ `  R )  =  |^| { x  |  ( R 
 C_  x  /\  ( x  o.  x )  C_  x ) } )
 
Theorembrintclab 12839* Two ways to express a binary relation which is the intersection of a class. (Contributed by RP, 4-Apr-2020.)
 |-  ( A |^| { x  |  ph } B  <->  A. x ( ph  -> 
 <. A ,  B >.  e.  x ) )
 
Theorembrtrclfv 12840* Two ways of expressing the transitive closure of a binary relation. (Contributed by RP, 9-May-2020.)
 |-  ( R  e.  V  ->  ( A ( t+ `  R ) B  <->  A. r ( ( R  C_  r  /\  ( r  o.  r
 )  C_  r )  ->  A r B ) ) )
 
Theorembrcnvtrclfv 12841* Two ways of expressing the transitive closure of the converse of a binary relation. (Contributed by RP, 9-May-2020.)
 |-  ( ( R  e.  U  /\  A  e.  V  /\  B  e.  W ) 
 ->  ( A `' (
 t+ `  R ) B  <->  A. r ( ( R  C_  r  /\  ( r  o.  r
 )  C_  r )  ->  B r A ) ) )
 
Theorembrtrclfvcnv 12842* Two ways of expressing the transitive closure of the converse of a binary relation. (Contributed by RP, 10-May-2020.)
 |-  ( R  e.  V  ->  ( A ( t+ `  `' R ) B  <->  A. r ( ( `' R  C_  r  /\  ( r  o.  r
 )  C_  r )  ->  A r B ) ) )
 
Theorembrcnvtrclfvcnv 12843* Two ways of expressing the transitive closure of the converse of the converse of a binary relation. (Contributed by RP, 10-May-2020.)
 |-  ( ( R  e.  U  /\  A  e.  V  /\  B  e.  W ) 
 ->  ( A `' (
 t+ `  `' R ) B  <->  A. r ( ( `' R  C_  r  /\  ( r  o.  r
 )  C_  r )  ->  B r A ) ) )
 
Theoremtrclfvss 12844 The transitive closure (as a relation) of a subclass is a subclass of the transitive closure. (Contributed by RP, 3-May-2020.)
 |-  ( ( R  e.  V  /\  S  e.  W  /\  R  C_  S )  ->  ( t+ `  R )  C_  ( t+ `  S ) )
 
Theoremtrclfvub 12845 The transitive closure of a relation has an upper bound. (Contributed by RP, 28-Apr-2020.)
 |-  ( R  e.  V  ->  ( t+ `  R )  C_  ( R  u.  ( dom  R  X.  ran  R ) ) )
 
Theoremtrclfvlb 12846 The transitive closure of a relation has a lower bound. (Contributed by RP, 28-Apr-2020.)
 |-  ( R  e.  V  ->  R  C_  ( t+ `  R )
 )
 
Theoremtrclfvcotr 12847 The transitive closure of a relation is a transitive relation. (Contributed by RP, 29-Apr-2020.)
 |-  ( R  e.  V  ->  ( ( t+ `
  R )  o.  ( t+ `  R ) )  C_  ( t+ `  R ) )
 
Theoremtrclfvlb2 12848 The transitive closure of a relation has a lower bound. (Contributed by RP, 8-May-2020.)
 |-  ( R  e.  V  ->  ( R  o.  R )  C_  ( t+ `
  R ) )
 
Theoremtrclfvlb3 12849 The transitive closure of a relation has a lower bound. (Contributed by RP, 8-May-2020.)
 |-  ( R  e.  V  ->  ( R  u.  ( R  o.  R ) ) 
 C_  ( t+ `
  R ) )
 
Theoremcotrtrclfv 12850 The transitive closure of a transitive relation. (Contributed by RP, 28-Apr-2020.)
 |-  ( ( R  e.  V  /\  ( R  o.  R )  C_  R ) 
 ->  ( t+ `  R )  =  R )
 
Theoremtrclidm 12851 The transitive closure of a relation is idempotent. (Contributed by RP, 29-Apr-2020.)
 |-  ( R  e.  V  ->  ( t+ `  ( t+ `  R ) )  =  ( t+ `  R ) )
 
Theoremtrclun 12852 Transitive closure of a union of relations. (Contributed by RP, 5-May-2020.)
 |-  ( ( R  e.  V  /\  S  e.  W )  ->  ( t+ `
  ( R  u.  S ) )  =  ( t+ `  ( ( t+ `
  R )  u.  ( t+ `  S ) ) ) )
 
Theoremtrclfvg 12853 The value of the transitive closure of a relation is a superset or (for proper classes) the empty set. (Contributed by RP, 8-May-2020.)
 |-  ( R  C_  (
 t+ `  R )  \/  ( t+ `
  R )  =  (/) )
 
Theoremtrclfvcotrg 12854 The value of the transitive closure of a relation is always a transitive relation. (Contributed by RP, 8-May-2020.)
 |-  ( ( t+ `
  R )  o.  ( t+ `  R ) )  C_  ( t+ `  R )
 
Theoremreltrclfv 12855 The transitive closure of a relation is a relation. (Contributed by RP, 9-May-2020.)
 |-  ( ( R  e.  V  /\  Rel  R )  ->  Rel  ( t+ `
  R ) )
 
Theoremdmtrclfv 12856 The domain of the transitive closure is equal to the domain of the relation. (Contributed by RP, 9-May-2020.)
 |-  ( R  e.  V  ->  dom  ( t+ `
  R )  = 
 dom  R )
 
5.8.4  Exponentiation of relations
 
Syntaxcrelexp 12857 Extend class notation to include relation exponentiation.
 class ^r
 
Definitiondf-relexp 12858* Definition of repeated composition of a relation with itself, aka relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 22-May-2020.)
 |- ^r  =  (
 r  e.  _V ,  n  e.  NN0  |->  if ( n  =  0 ,  (  _I  |`  ( dom  r  u.  ran  r )
 ) ,  (  seq 1 ( ( x  e.  _V ,  y  e.  _V  |->  ( x  o.  r ) ) ,  ( z  e.  _V  |->  r ) ) `  n ) ) )
 
Theoremrelexp0g 12859 A relation composed zero times is the (restricted) identity. (Contributed by RP, 22-May-2020.)
 |-  ( R  e.  V  ->  ( R ^r 
 0 )  =  (  _I  |`  ( dom  R  u.  ran  R )
 ) )
 
Theoremrelexp0 12860 A relation composed zero times is the (restricted) identity. (Contributed by RP, 22-May-2020.)
 |-  ( ( R  e.  V  /\  Rel  R )  ->  ( R ^r 
 0 )  =  (  _I  |`  U. U. R ) )
 
Theoremrelexp0d 12861 A relation composed zero times is the (restricted) identity. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( R ^r  0 )  =  (  _I  |`  U. U. R ) )
 
Theoremrelexpsucnnr 12862 A reduction for relation exponentiation to the right. (Contributed by RP, 22-May-2020.)
 |-  ( ( R  e.  V  /\  N  e.  NN )  ->  ( R ^r  ( N  +  1 ) )  =  ( ( R ^r  N )  o.  R ) )
 
Theoremrelexp1g 12863 A relation composed once is itself. (Contributed by RP, 22-May-2020.)
 |-  ( R  e.  V  ->  ( R ^r 
 1 )  =  R )
 
Theoremrelexpsucr 12864 A reduction for relation exponentiation to the right. (Contributed by RP, 23-May-2020.)
 |-  ( ( R  e.  V  /\  Rel  R  /\  N  e.  NN0 )  ->  ( R ^r 
 ( N  +  1 ) )  =  ( ( R ^r  N )  o.  R ) )
 
Theoremrelexpsucrd 12865 A reduction for relation exponentiation to the right. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  ( R ^r  ( N  +  1 ) )  =  ( ( R ^r  N )  o.  R ) ) )
 
Theoremrelexp1d 12866 A relation composed once is itself. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( R ^r  1 )  =  R )
 
Theoremrelexpsucnnl 12867 A reduction for relation exponentiation to the left. (Contributed by RP, 23-May-2020.)
 |-  ( ( R  e.  V  /\  N  e.  NN )  ->  ( R ^r  ( N  +  1 ) )  =  ( R  o.  ( R ^r  N ) ) )
 
Theoremrelexpsucl 12868 A reduction for relation exponentiation to the left. (Contributed by RP, 23-May-2020.)
 |-  ( ( R  e.  V  /\  Rel  R  /\  N  e.  NN0 )  ->  ( R ^r 
 ( N  +  1 ) )  =  ( R  o.  ( R ^r  N ) ) )
 
Theoremrelexpsucld 12869 A reduction for relation exponentiation to the left. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  ( R ^r  ( N  +  1 ) )  =  ( R  o.  ( R ^r  N ) ) ) )
 
Theoremrelexpcnv 12870 Commutation of converse and relation exponentiation. (Contributed by RP, 23-May-2020.)
 |-  ( ( N  e.  NN0  /\  R  e.  V ) 
 ->  `' ( R ^r  N )  =  ( `' R ^r  N ) )
 
Theoremrelexpcnvd 12871 Commutation of converse and relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  `' ( R ^r  N )  =  ( `' R ^r  N ) ) )
 
Theoremrelexp0rel 12872 The exponentiation of a class to zero is a relation. (Contributed by RP, 23-May-2020.)
 |-  ( R  e.  V  ->  Rel  ( R ^r  0 ) )
 
Theoremrelexprelg 12873 The exponentiation of a class is a relation except when the exponent is one and the class is not a relation. (Contributed by RP, 23-May-2020.)
 |-  ( ( N  e.  NN0  /\  R  e.  V  /\  ( N  =  1  ->  Rel  R ) ) 
 ->  Rel  ( R ^r  N ) )
 
Theoremrelexprel 12874 The exponentiation of a relation is a relation. (Contributed by RP, 23-May-2020.)
 |-  ( ( N  e.  NN0  /\  R  e.  V  /\  Rel 
 R )  ->  Rel  ( R ^r  N ) )
 
Theoremrelexpreld 12875 The exponentiation of a relation is a relation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  Rel  ( R ^r  N ) ) )
 
Theoremrelexpnndm 12876 The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
 |-  ( ( N  e.  NN  /\  R  e.  V )  ->  dom  ( R ^r  N )  C_  dom 
 R )
 
Theoremrelexpdmg 12877 The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
 |-  ( ( N  e.  NN0  /\  R  e.  V ) 
 ->  dom  ( R ^r  N )  C_  ( dom  R  u.  ran  R ) )
 
Theoremrelexpdm 12878 The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
 |-  ( ( N  e.  NN0  /\  R  e.  V ) 
 ->  dom  ( R ^r  N )  C_  U. U. R )
 
Theoremrelexpdmd 12879 The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  dom  ( R ^r  N ) 
 C_  U. U. R ) )
 
Theoremrelexpnnrn 12880 The range of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
 |-  ( ( N  e.  NN  /\  R  e.  V )  ->  ran  ( R ^r  N )  C_  ran 
 R )
 
Theoremrelexprng 12881 The range of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
 |-  ( ( N  e.  NN0  /\  R  e.  V ) 
 ->  ran  ( R ^r  N )  C_  ( dom  R  u.  ran  R ) )
 
Theoremrelexprn 12882 The range of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
 |-  ( ( N  e.  NN0  /\  R  e.  V ) 
 ->  ran  ( R ^r  N )  C_  U. U. R )
 
Theoremrelexprnd 12883 The range of an exponentiation of a relation a subset of the relation's field. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  ran  ( R ^r  N ) 
 C_  U. U. R ) )
 
Theoremrelexpfld 12884 The field of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
 |-  ( ( N  e.  NN0  /\  R  e.  V ) 
 ->  U. U. ( R ^r  N ) 
 C_  U. U. R )
 
Theoremrelexpfldd 12885 The field of an exponentiation of a relation a subset of the relation's field. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  U. U. ( R ^r  N )  C_  U. U. R ) )
 
Theoremrelexpaddnn 12886 Relation composition becomes addition under exponentiation. (Contributed by RP, 23-May-2020.)
 |-  ( ( N  e.  NN  /\  M  e.  NN  /\  R  e.  V ) 
 ->  ( ( R ^r  N )  o.  ( R ^r  M ) )  =  ( R ^r  ( N  +  M ) ) )
 
Theoremrelexpuzrel 12887 The exponentiation of a class to an integer not smaller than 2 is a relation. (Contributed by RP, 23-May-2020.)
 |-  ( ( N  e.  ( ZZ>= `  2 )  /\  R  e.  V ) 
 ->  Rel  ( R ^r  N ) )
 
Theoremrelexpaddg 12888 Relation composition becomes addition under exponentiation except when the exponents total to one and the class isn't a relation. (Contributed by RP, 30-May-2020.)
 |-  ( ( N  e.  NN0  /\  ( M  e.  NN0  /\  R  e.  V  /\  ( ( N  +  M )  =  1  ->  Rel  R ) ) )  ->  ( ( R ^r  N )  o.  ( R ^r  M ) )  =  ( R ^r 
 ( N  +  M ) ) )
 
Theoremrelexpaddd 12889 Relation composition becomes addition under exponentiation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  (
 ( N  e.  NN0  /\  M  e.  NN0 )  ->  ( ( R ^r  N )  o.  ( R ^r  M ) )  =  ( R ^r  ( N  +  M ) ) ) )
 
5.8.5  Reflexive-transitive closure as an indexed union
 
Syntaxcrtrcl 12890 Extend class notation with recursively defined reflexive, transitive closure.
 class 
 t*rec
 
Definitiondf-rtrclrec 12891* The reflexive, transitive closure of a relation constructed as the union of all finite exponentiations. (Contributed by Drahflow, 12-Nov-2015.)
 |-  t*rec  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^r  n ) )
 
Theoremdfrtrclrec2 12892* If two elements are connected by a reflexive, transitive closure, then they are connected via  n instances the relation, for some  n. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( A ( t*rec
 `  R ) B  <->  E. n  e.  NN0  A ( R ^r  n ) B ) )
 
Theoremrtrclreclem1 12893 The reflexive, transitive closure is indeed reflexive. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  (  _I  |`  U. U. R )  C_  ( t*rec
 `  R ) )
 
Theoremrtrclreclem2 12894 The reflexive, transitive closure is indeed a closure. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  R  C_  ( t*rec `  R ) )
 
Theoremrtrclreclem3 12895 The reflexive, transitive closure is indeed transitive. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  (
 ( t*rec `  R )  o.  (
 t*rec `  R ) )  C_  ( t*rec `  R )
 )
 
Theoremrtrclreclem4 12896* The reflexive, transitive closure of  R is the smallest reflexive, transitive relation which contains  R and the identity. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  A. s
 ( ( (  _I  |`  ( dom  R  u.  ran 
 R ) )  C_  s  /\  R  C_  s  /\  ( s  o.  s
 )  C_  s )  ->  ( t*rec `  R )  C_  s ) )
 
Theoremdfrtrcl2 12897 The two definitions  t* and  t*rec of the reflexive, transitive closure coincide if  R is indeed a relation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  (
 t* `  R )  =  ( t*rec `  R ) )
 
5.8.6  Principle of transitive induction.

If we have a statement that holds for some element, and a relation between elements that implies if it holds for the first element then it must hold for the second element, the principle of transitive induction shows the statement holds for any element related to the first by the (reflexive-)transitive closure of the relation.

 
Theoremrelexpindlem 12898* Principle of transitive induction, finite and non-class version. The first three hypotheses give various existences, the next three give necessary substitutions and the last two are the basis and the induction hypothesis. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
 |-  ( et  ->  Rel  R )   &    |-  ( et  ->  R  e.  _V )   &    |-  ( et  ->  S  e.  _V )   &    |-  (
 i  =  S  ->  (
 ph 
 <->  ch ) )   &    |-  (
 i  =  x  ->  ( ph  <->  ps ) )   &    |-  (
 i  =  j  ->  ( ph  <->  th ) )   &    |-  ( et  ->  ch )   &    |-  ( et  ->  ( j R x  ->  ( th  ->  ps )
 ) )   =>    |-  ( et  ->  ( n  e.  NN0  ->  ( S ( R ^r  n ) x  ->  ps ) ) )
 
Theoremrelexpind 12899* Principle of transitive induction, finite version. The first three hypotheses give various existences, the next three give necessary substitutions and the last two are the basis and the induction hypothesis. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
 |-  ( et  ->  Rel  R )   &    |-  ( et  ->  R  e.  _V )   &    |-  ( et  ->  S  e.  _V )   &    |-  ( et  ->  X  e.  _V )   &    |-  ( i  =  S  ->  ( ph  <->  ch ) )   &    |-  (
 i  =  x  ->  ( ph  <->  ps ) )   &    |-  (
 i  =  j  ->  ( ph  <->  th ) )   &    |-  ( x  =  X  ->  ( ps  <->  ta ) )   &    |-  ( et  ->  ch )   &    |-  ( et  ->  ( j R x  ->  ( th  ->  ps )
 ) )   =>    |-  ( et  ->  ( n  e.  NN0  ->  ( S ( R ^r  n ) X  ->  ta ) ) )
 
Theoremrtrclind 12900* Principle of transitive induction. The first four hypotheses give various existences, the next four give necessary substitutions and the last two are the basis and the induction step. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( et  ->  Rel  R )   &    |-  ( et  ->  R  e.  _V )   &    |-  ( et  ->  S  e.  _V )   &    |-  ( et  ->  X  e.  _V )   &    |-  ( i  =  S  ->  ( ph  <->  ch ) )   &    |-  (
 i  =  x  ->  ( ph  <->  ps ) )   &    |-  (
 i  =  j  ->  ( ph  <->  th ) )   &    |-  ( x  =  X  ->  ( ps  <->  ta ) )   &    |-  ( et  ->  ch )   &    |-  ( et  ->  ( j R x  ->  ( th  ->  ps )
 ) )   =>    |-  ( et  ->  ( S ( t* `
  R ) X 
 ->  ta ) )
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