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Theorem List for Metamath Proof Explorer - 36301-36400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremclcnvlem 36301* When  A, an upper bound of the closure, exists and certain substitutions hold the converse of the closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
 |-  (
 ( ph  /\  x  =  ( `' y  u.  ( X  \  `' `' X ) ) ) 
 ->  ( ch  ->  ps )
 )   &    |-  ( ( ph  /\  y  =  `' x )  ->  ( ps  ->  ch ) )   &    |-  ( x  =  A  ->  ( ps  <->  th ) )   &    |-  ( ph  ->  X  C_  A )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  th )   =>    |-  ( ph  ->  `' |^| { x  |  ( X 
 C_  x  /\  ps ) }  =  |^| { y  |  ( `' X  C_  y  /\  ch ) } )
 
Theoremcnvtrucl0 36302* The converse of the trivial closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
 |-  ( X  e.  V  ->  `'
 |^| { x  |  ( X  C_  x  /\ T.  ) }  =  |^| { y  |  ( `' X  C_  y  /\ T.  ) } )
 
Theoremcnvrcl0 36303* The converse of the reflexive closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
 |-  ( X  e.  V  ->  `'
 |^| { x  |  ( X  C_  x  /\  (  _I  |`  ( dom  x  u.  ran  x )
 )  C_  x ) }  =  |^| { y  |  ( `' X  C_  y  /\  (  _I  |`  ( dom  y  u.  ran  y
 ) )  C_  y
 ) } )
 
Theoremcnvtrcl0 36304* The converse of the transitive closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
 |-  ( X  e.  V  ->  `'
 |^| { x  |  ( X  C_  x  /\  ( x  o.  x )  C_  x ) }  =  |^| { y  |  ( `' X  C_  y  /\  ( y  o.  y )  C_  y
 ) } )
 
Theoremdmtrcl 36305* The domain of the transitive closure is equal to the domain of its base relation. (Contributed by RP, 1-Nov-2020.)
 |-  ( X  e.  V  ->  dom  |^| { x  |  ( X  C_  x  /\  ( x  o.  x )  C_  x ) }  =  dom  X )
 
Theoremrntrcl 36306* The range of the transitive closure is equal to the range of its base relation. (Contributed by RP, 1-Nov-2020.)
 |-  ( X  e.  V  ->  ran  |^| { x  |  ( X  C_  x  /\  ( x  o.  x )  C_  x ) }  =  ran  X )
 
Theoremdfrtrcl5 36307* Definition of reflexive-transitive closure as a standard closure. (Contributed by RP, 1-Nov-2020.)
 |-  t*  =  ( x  e.  _V  |->  |^| { y  |  ( x  C_  y  /\  ( (  _I  |`  ( dom  y  u.  ran  y
 ) )  C_  y  /\  ( y  o.  y
 )  C_  y )
 ) } )
 
21.25.1.16  RP REPLACE: Definitions and basic properties of transitive closures
 
Theoremtrcleq2lemRP 36308 Equality implies bijection. (Contributed by RP, 5-May-2020.) (Proof modification is discouraged.)
 |-  ( A  =  B  ->  ( ( R  C_  A  /\  ( A  o.  A )  C_  A )  <->  ( R  C_  B  /\  ( B  o.  B )  C_  B ) ) )
 
21.25.2  Additional statements on relations and subclasses
 
Theoremal3im 36309 Version of ax-4 1690 for a nested implication. (Contributed by RP, 13-Apr-2020.)
 |-  ( A. x ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) 
 ->  ( A. x ph  ->  ( A. x ps  ->  ( A. x ch  ->  A. x th )
 ) ) )
 
Theoremintima0 36310* Two ways of expressing the intersection of images of a class. (Contributed by RP, 13-Apr-2020.)
 |-  |^|_ a  e.  A  ( a " B )  =  |^| { x  |  E. a  e.  A  x  =  ( a " B ) }
 
Theoremelimaint 36311* Element of image of intersection. (Contributed by RP, 13-Apr-2020.)
 |-  (
 y  e.  ( |^| A
 " B )  <->  E. b  e.  B  A. a  e.  A  <. b ,  y >.  e.  a
 )
 
Theoremcsbcog 36312 Distribute proper substitution through a composition of relations. (Contributed by RP, 28-Jun-2020.)
 |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B  o.  C )  =  ( [_ A  /  x ]_ B  o.  [_ A  /  x ]_ C ) )
 
Theoremcnviun 36313* Converse of indexed union. (Contributed by RP, 20-Jun-2020.)
 |-  `' U_ x  e.  A  B  =  U_ x  e.  A  `' B
 
Theoremimaiun1 36314* The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020.)
 |-  ( U_ x  e.  A  B " C )  = 
 U_ x  e.  A  ( B " C )
 
Theoremcoiun1 36315* Composition with an indexed union. Proof analgous to that of coiun 5352. (Contributed by RP, 20-Jun-2020.)
 |-  ( U_ x  e.  C  A  o.  B )  = 
 U_ x  e.  C  ( A  o.  B )
 
Theoremelintima 36316* Element of intersection of images. (Contributed by RP, 13-Apr-2020.)
 |-  (
 y  e.  |^| { x  |  E. a  e.  A  x  =  ( a " B ) }  <->  A. a  e.  A  E. b  e.  B  <. b ,  y >.  e.  a )
 
Theoremintimass 36317* The image under the intersection of relations is a subset of the intersection of the images. (Contributed by RP, 13-Apr-2020.)
 |-  ( |^| A " B ) 
 C_  |^| { x  |  E. a  e.  A  x  =  ( a " B ) }
 
Theoremintimass2 36318* The image under the intersection of relations is a subset of the intersection of the images. (Contributed by RP, 13-Apr-2020.)
 |-  ( |^| A " B ) 
 C_  |^|_ x  e.  A  ( x " B )
 
Theoremintimag 36319* Requirement for the image under the intersection of relations to equal the intersection of the images of those relations. (Contributed by RP, 13-Apr-2020.)
 |-  ( A. y ( A. a  e.  A  E. b  e.  B  <. b ,  y >.  e.  a  ->  E. b  e.  B  A. a  e.  A  <. b ,  y >.  e.  a )  ->  ( |^| A " B )  =  |^| { x  |  E. a  e.  A  x  =  ( a " B ) } )
 
Theoremintimasn 36320* Two ways to express the image of a singleton when the relation is an intersection. (Contributed by RP, 13-Apr-2020.)
 |-  ( B  e.  V  ->  (
 |^| A " { B } )  =  |^| { x  |  E. a  e.  A  x  =  ( a " { B } ) } )
 
Theoremintimasn2 36321* Two ways to express the image of a singleton when the relation is an intersection. (Contributed by RP, 13-Apr-2020.)
 |-  ( B  e.  V  ->  (
 |^| A " { B } )  =  |^|_ x  e.  A  ( x
 " { B }
 ) )
 
Theoremss2iundf 36322* Subclass theorem for indexed union. (Contributed by RP, 17-Jul-2020.)
 |-  F/ x ph   &    |-  F/ y ph   &    |-  F/_ y Y   &    |-  F/_ y A   &    |-  F/_ y B   &    |-  F/_ x C   &    |-  F/_ y C   &    |-  F/_ x D   &    |-  F/_ y G   &    |-  ( ( ph  /\  x  e.  A )  ->  Y  e.  C )   &    |-  ( ( ph  /\  x  e.  A  /\  y  =  Y )  ->  D  =  G )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  C_  G )   =>    |-  ( ph  ->  U_ x  e.  A  B  C_  U_ y  e.  C  D )
 
Theoremss2iundv 36323* Subclass theorem for indexed union. (Contributed by RP, 17-Jul-2020.)
 |-  (
 ( ph  /\  x  e.  A )  ->  Y  e.  C )   &    |-  ( ( ph  /\  x  e.  A  /\  y  =  Y )  ->  D  =  G )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  C_  G )   =>    |-  ( ph  ->  U_ x  e.  A  B  C_  U_ y  e.  C  D )
 
Theoremcbviuneq12df 36324* Rule used to change the bound variables and classes in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by RP, 17-Jul-2020.)
 |-  F/ x ph   &    |-  F/ y ph   &    |-  F/_ x X   &    |-  F/_ y Y   &    |-  F/_ x A   &    |-  F/_ y A   &    |-  F/_ y B   &    |-  F/_ x C   &    |-  F/_ y C   &    |-  F/_ x D   &    |-  F/_ x F   &    |-  F/_ y G   &    |-  ( ( ph  /\  y  e.  C )  ->  X  e.  A )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  Y  e.  C )   &    |-  ( ( ph  /\  y  e.  C  /\  x  =  X )  ->  B  =  F )   &    |-  ( ( ph  /\  x  e.  A  /\  y  =  Y )  ->  D  =  G )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  =  G )   &    |-  ( ( ph  /\  y  e.  C ) 
 ->  D  =  F )   =>    |-  ( ph  ->  U_ x  e.  A  B  =  U_ y  e.  C  D )
 
Theoremcbviuneq12dv 36325* Rule used to change the bound variables and classes in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by RP, 17-Jul-2020.)
 |-  (
 ( ph  /\  y  e.  C )  ->  X  e.  A )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  Y  e.  C )   &    |-  ( ( ph  /\  y  e.  C  /\  x  =  X )  ->  B  =  F )   &    |-  ( ( ph  /\  x  e.  A  /\  y  =  Y )  ->  D  =  G )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  =  G )   &    |-  ( ( ph  /\  y  e.  C ) 
 ->  D  =  F )   =>    |-  ( ph  ->  U_ x  e.  A  B  =  U_ y  e.  C  D )
 
Theoremconrel1d 36326 Deduction about composition with a class with no relational content. (Contributed by Richard Penner, 24-Dec-2019.)
 |-  ( ph  ->  `' A  =  (/) )   =>    |-  ( ph  ->  ( A  o.  B )  =  (/) )
 
Theoremconrel2d 36327 Deduction about composition with a class with no relational content. (Contributed by Richard Penner, 24-Dec-2019.)
 |-  ( ph  ->  `' A  =  (/) )   =>    |-  ( ph  ->  ( B  o.  A )  =  (/) )
 
21.25.2.1  Transitive relations (not to be confused with transitive classes).
 
Theoremtrrelind 36328 The intersection of transitive relations is a transitive relation. (Contributed by Richard Penner, 24-Dec-2019.)
 |-  ( ph  ->  ( R  o.  R )  C_  R )   &    |-  ( ph  ->  ( S  o.  S )  C_  S )   &    |-  ( ph  ->  T  =  ( R  i^i  S ) )   =>    |-  ( ph  ->  ( T  o.  T )  C_  T )
 
Theoremxpintrreld 36329 The intersection of a transitive relation with a cross product is a transitve relation. (Contributed by Richard Penner, 24-Dec-2019.)
 |-  ( ph  ->  ( R  o.  R )  C_  R )   &    |-  ( ph  ->  S  =  ( R  i^i  ( A  X.  B ) ) )   =>    |-  ( ph  ->  ( S  o.  S )  C_  S )
 
Theoremrestrreld 36330 The restriction of a transitive relation is a transitive relation. (Contributed by Richard Penner, 24-Dec-2019.)
 |-  ( ph  ->  ( R  o.  R )  C_  R )   &    |-  ( ph  ->  S  =  ( R  |`  A ) )   =>    |-  ( ph  ->  ( S  o.  S )  C_  S )
 
Theoremtrrelsuperreldg 36331 Concrete construction of a superclass of relation  R which is a transitive relation. (Contributed by Richard Penner, 25-Dec-2019.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  S  =  ( dom  R  X.  ran  R ) )   =>    |-  ( ph  ->  ( R  C_  S  /\  ( S  o.  S )  C_  S ) )
 
Theoremtrficl 36332* The class of all transitive relations has the finite intersection property. (Contributed by Richard Penner, 1-Jan-2020.) (Proof shortened by Richard Penner, 3-Jan-2020.)
 |-  A  =  { z  |  ( z  o.  z ) 
 C_  z }   =>    |-  A. x  e.  A  A. y  e.  A  ( x  i^i  y )  e.  A
 
Theoremcnvtrrel 36333 The converse of a transitive relation is a transitive relation. (Contributed by Richard Penner, 25-Dec-2019.)
 |-  (
 ( S  o.  S )  C_  S  <->  ( `' S  o.  `' S )  C_  `' S )
 
Theoremtrrelsuperrel2dg 36334 Concrete construction of a superclass of relation  R which is a transitive relation. (Contributed by RP, 20-Jul-2020.)
 |-  ( ph  ->  S  =  ( R  u.  ( dom 
 R  X.  ran  R ) ) )   =>    |-  ( ph  ->  ( R  C_  S  /\  ( S  o.  S )  C_  S ) )
 
21.25.2.2  Reflexive closures
 
Syntaxcrcl 36335 Extend class notation with reflexive closure.
 class  r*
 
Definitiondf-rcl 36336* Reflexive closure of a relation. This is the smallest superset which has the reflexive property. (Contributed by RP, 5-Jun-2020.)
 |-  r*  =  ( x  e.  _V  |->  |^| { z  |  ( x  C_  z  /\  (  _I  |`  ( dom  z  u.  ran  z
 ) )  C_  z
 ) } )
 
Theoremdfrcl2 36337 Reflexive closure of a relation as union with restricted identity relation. (Contributed by RP, 6-Jun-2020.)
 |-  r*  =  ( x  e.  _V  |->  ( (  _I  |`  ( dom  x  u.  ran 
 x ) )  u.  x ) )
 
Theoremdfrcl3 36338 Reflexive closure of a relation as union of powers of the relation. (Contributed by RP, 6-Jun-2020.)
 |-  r*  =  ( x  e.  _V  |->  ( ( x ^r  0 )  u.  ( x ^r  1 ) ) )
 
Theoremdfrcl4 36339* Reflexive closure of a relation as indexed union of powers of the relation. (Contributed by RP, 8-Jun-2020.)
 |-  r*  =  ( r  e.  _V  |->  U_ n  e.  {
 0 ,  1 }  ( r ^r  n ) )
 
21.25.2.3  Finite relationship composition.

In order for theorems on the transitive closure of a relation to be grouped together before the concept of continuity, we really need an analogue of ^r that works on finite ordinals or finite sets instead of natural numbers.

 
Theoremrelexp2 36340 A set operated on by the relation exponent to the second power is equal to the composition of the set with itself. (Contributed by RP, 1-Jun-2020.)
 |-  ( R  e.  V  ->  ( R ^r  2 )  =  ( R  o.  R ) )
 
Theoremrelexpnul 36341 If the domain and range of powers of a relation are disjoint then the relation raised to the sum of those exponents is empty. (Contributed by RP, 1-Jun-2020.)
 |-  (
 ( ( R  e.  V  /\  Rel  R )  /\  ( N  e.  NN0  /\  M  e.  NN0 )
 )  ->  ( ( dom  ( R ^r  N )  i^i  ran  ( R ^r  M ) )  =  (/)  <->  ( R ^r  ( N  +  M ) )  =  (/) ) )
 
Theoremeliunov2 36342* Membership in the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the the element is a member of that operator value. Generalized from dfrtrclrec2 13197. (Contributed by RP, 1-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  N  ( r  .^  n )
 )   =>    |-  ( ( R  e.  U  /\  N  e.  V )  ->  ( X  e.  ( C `  R )  <->  E. n  e.  N  X  e.  ( R  .^  n ) ) )
 
Theoremeltrclrec 36343* Membership in the indexed union of relation exponentiation over the natural numbers is equivalent to the existence of at least one number such that the element is a member of that relationship power. (Contributed by RP, 2-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  NN  (
 r ^r  n ) )   =>    |-  ( R  e.  V  ->  ( X  e.  ( C `  R )  <->  E. n  e.  NN  X  e.  ( R ^r  n ) ) )
 
Theoremelrtrclrec 36344* Membership in the indexed union of relation exponentiation over the natural numbers (including zero) is equivalent to the existence of at least one number such that the element is a member of that relationship power. (Contributed by RP, 2-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  NN0  (
 r ^r  n ) )   =>    |-  ( R  e.  V  ->  ( X  e.  ( C `  R )  <->  E. n  e.  NN0  X  e.  ( R ^r  n ) ) )
 
Theorembriunov2 36345* Two classes related by the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the the two classes are related by that operator value. (Contributed by RP, 1-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  N  ( r  .^  n )
 )   =>    |-  ( ( R  e.  U  /\  N  e.  V )  ->  ( X ( C `  R ) Y  <->  E. n  e.  N  X ( R  .^  n ) Y ) )
 
Theorembrmptiunrelexpd 36346* If two elements are connected by an indexed union of relational powers, then they are connected via 
n instances the relation, for some  n. Generalization of dfrtrclrec2 13197. (Contributed by RP, 21-Jul-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  N  ( r ^r  n ) )   &    |-  ( ph  ->  R  e.  _V )   &    |-  ( ph  ->  N  C_  NN0 )   =>    |-  ( ph  ->  ( A ( C `  R ) B  <->  E. n  e.  N  A ( R ^r  n ) B ) )
 
Theoremfvmptiunrelexplb0d 36347* If the indexed union ranges over the zeroth power of the relation, then a restriction of the identity relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  N  ( r ^r  n ) )   &    |-  ( ph  ->  R  e.  _V )   &    |-  ( ph  ->  N  e.  _V )   &    |-  ( ph  ->  0  e.  N )   =>    |-  ( ph  ->  (  _I  |`  ( dom  R  u.  ran  R ) ) 
 C_  ( C `  R ) )
 
Theoremfvmptiunrelexplb0da 36348* If the indexed union ranges over the zeroth power of the relation, then a restriction of the identity relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  N  ( r ^r  n ) )   &    |-  ( ph  ->  R  e.  _V )   &    |-  ( ph  ->  N  e.  _V )   &    |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  0  e.  N )   =>    |-  ( ph  ->  (  _I  |`  U. U. R )  C_  ( C `  R ) )
 
Theoremfvmptiunrelexplb1d 36349* If the indexed union ranges over the first power of the relation, then the relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  N  ( r ^r  n ) )   &    |-  ( ph  ->  R  e.  _V )   &    |-  ( ph  ->  N  e.  _V )   &    |-  ( ph  ->  1  e.  N )   =>    |-  ( ph  ->  R  C_  ( C `  R ) )
 
Theorembrfvid 36350 If two elements are connected by a value of the identity relation, then they are connected via the argument. (Contributed by RP, 21-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( A (  _I  `  R ) B  <->  A R B ) )
 
TheorembrfvidRP 36351 If two elements are connected by a value of the identity relation, then they are connected via the argument. This is an example which uses brmptiunrelexpd 36346. (Contributed by RP, 21-Jul-2020.) (Proof modification is discouraged.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( A (  _I  `  R ) B  <->  A R B ) )
 
Theoremfvilbd 36352 A set is a subset of its image under the identity relation. (Contributed by RP, 22-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  R  C_  (  _I  `  R ) )
 
TheoremfvilbdRP 36353 A set is a subset of its image under the identity relation. (Contributed by RP, 22-Jul-2020.) (Proof modification is discouraged.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  R  C_  (  _I  `  R ) )
 
Theorembrfvrcld 36354 If two elements are connected by the reflexive closure of a relation, then they are connected via zero or one instances the relation. (Contributed by RP, 21-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( A ( r* `
  R ) B  <-> 
 ( A ( R ^r  0 ) B  \/  A ( R ^r  1 ) B ) ) )
 
Theorembrfvrcld2 36355 If two elements are connected by the reflexive closure of a relation, then they are equal or related by relation. (Contributed by RP, 21-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( A ( r* `
  R ) B  <-> 
 ( ( A  e.  ( dom  R  u.  ran  R )  /\  B  e.  ( dom  R  u.  ran  R )  /\  A  =  B )  \/  A R B ) ) )
 
Theoremfvrcllb0d 36356 A restriction of the identity relation is a subset of the reflexive closure of a set. (Contributed by RP, 22-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  (  _I  |`  ( dom  R  u.  ran  R ) ) 
 C_  ( r* `
  R ) )
 
Theoremfvrcllb0da 36357 A restriction of the identity relation is a subset of the reflexive closure of a relation. (Contributed by RP, 22-Jul-2020.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  (  _I  |`  U. U. R )  C_  ( r* `
  R ) )
 
Theoremfvrcllb1d 36358 A set is a subset of its image under the reflexive closure. (Contributed by RP, 22-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  R  C_  ( r* `  R ) )
 
Theorembrtrclrec 36359* Two classes related by the indexed union of relation exponentiation over the natural numbers is equivalent to the existence of at least one number such that the two classes are related by that relationship power. (Contributed by RP, 2-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  NN  (
 r ^r  n ) )   =>    |-  ( R  e.  V  ->  ( X ( C `
  R ) Y  <->  E. n  e.  NN  X ( R ^r  n ) Y ) )
 
Theorembrrtrclrec 36360* Two classes related by the indexed union of relation exponentiation over the natural numbers (including zero) is equivalent to the existence of at least one number such that the two classes are related by that relationship power. (Contributed by RP, 2-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  NN0  (
 r ^r  n ) )   =>    |-  ( R  e.  V  ->  ( X ( C `
  R ) Y  <->  E. n  e.  NN0  X ( R ^r  n ) Y ) )
 
Theorembriunov2uz 36361* Two classes related by the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the the two classes are related by that operator value. The index set  N is restricted to an upper range of integers. (Contributed by RP, 2-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  N  ( r  .^  n )
 )   =>    |-  ( ( R  e.  U  /\  N  =  (
 ZZ>= `  M ) ) 
 ->  ( X ( C `
  R ) Y  <->  E. n  e.  N  X ( R  .^  n ) Y ) )
 
Theoremeliunov2uz 36362* Membership in the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the the element is a member of that operator value. The index set  N is restricted to an upper range of integers. (Contributed by RP, 2-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  N  ( r  .^  n )
 )   =>    |-  ( ( R  e.  U  /\  N  =  (
 ZZ>= `  M ) ) 
 ->  ( X  e.  ( C `  R )  <->  E. n  e.  N  X  e.  ( R  .^  n ) ) )
 
Theoremov2ssiunov2 36363* Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 13198 and rtrclreclem2 . (Contributed by RP, 4-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  N  ( r  .^  n )
 )   =>    |-  ( ( R  e.  U  /\  N  e.  V  /\  M  e.  N ) 
 ->  ( R  .^  M )  C_  ( C `  R ) )
 
Theoremrelexp0eq 36364 The zeroth power of relationships is the same if and only if the union of their domain and ranges is the same. (Contributed by RP, 11-Jun-2020.)
 |-  (
 ( A  e.  U  /\  B  e.  V ) 
 ->  ( ( dom  A  u.  ran  A )  =  ( dom  B  u.  ran 
 B )  <->  ( A ^r  0 )  =  ( B ^r 
 0 ) ) )
 
Theoremiunrelexp0 36365* Simplification of zeroth power of indexed union of powers of relations. (Contributed by RP, 19-Jun-2020.)
 |-  (
 ( R  e.  V  /\  Z  C_  NN0  /\  ( { 0 ,  1 }  i^i  Z )  =/=  (/) )  ->  ( U_ x  e.  Z  ( R ^r  x ) ^r 
 0 )  =  ( R ^r  0 ) )
 
Theoremrelexpxpnnidm 36366 Any positive power of a cross product of non-disjoint sets is itself. (Contributed by RP, 13-Jun-2020.)
 |-  ( N  e.  NN  ->  ( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) )  ->  (
 ( A  X.  B ) ^r  N )  =  ( A  X.  B ) ) )
 
Theoremrelexpiidm 36367 Any power of any restriction of the identity relation is itself. (Contributed by RP, 12-Jun-2020.)
 |-  (
 ( A  e.  V  /\  N  e.  NN0 )  ->  ( (  _I  |`  A ) ^r  N )  =  (  _I  |`  A ) )
 
Theoremrelexpss1d 36368 The relational power of a subset is a subset. (Contributed by RP, 17-Jun-2020.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( A ^r  N ) 
 C_  ( B ^r  N ) )
 
Theoremcomptiunov2i 36369* The composition two indexed unions is sometimes a similar indexed union. (Contributed by RP, 10-Jun-2020.)
 |-  X  =  ( a  e.  _V  |->  U_ i  e.  I  ( a  .^  i )
 )   &    |-  Y  =  ( b  e.  _V  |->  U_ j  e.  J  ( b  .^  j ) )   &    |-  Z  =  ( c  e.  _V  |->  U_ k  e.  K  ( c  .^  k )
 )   &    |-  I  e.  _V   &    |-  J  e.  _V   &    |-  K  =  ( I  u.  J )   &    |-  U_ k  e.  I  ( d  .^  k )  C_  U_ i  e.  I  ( U_ j  e.  J  ( d  .^  j ) 
 .^  i )   &    |-  U_ k  e.  J  ( d  .^  k )  C_  U_ i  e.  I  ( U_ j  e.  J  (
 d  .^  j )  .^  i )   &    |-  U_ i  e.  I  ( U_ j  e.  J  ( d  .^  j ) 
 .^  i )  C_  U_ k  e.  ( I  u.  J ) ( d  .^  k )   =>    |-  ( X  o.  Y )  =  Z
 
Theoremcorclrcl 36370 The reflexive closure is idempotent. (Contributed by RP, 13-Jun-2020.)
 |-  (
 r*  o.  r* )  =  r*
 
Theoremiunrelexpmin1 36371* The indexed union of relation exponentiation over the natural numbers is the minimum transitive relation that includes the relation. (Contributed by RP, 4-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  N  ( r ^r  n ) )   =>    |-  ( ( R  e.  V  /\  N  =  NN )  ->  A. s ( ( R  C_  s  /\  ( s  o.  s
 )  C_  s )  ->  ( C `  R )  C_  s ) )
 
Theoremrelexpmulnn 36372 With exponents limited to the counting numbers, the composition of powers of a relation is the relation raised to the product of exponents. (Contributed by RP, 13-Jun-2020.)
 |-  (
 ( ( R  e.  V  /\  I  =  ( J  x.  K ) )  /\  ( J  e.  NN  /\  K  e.  NN ) )  ->  ( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) )
 
Theoremrelexpmulg 36373 With ordered exponents, the composition of powers of a relation is the relation raised to the product of exponents. (Contributed by RP, 13-Jun-2020.)
 |-  (
 ( ( R  e.  V  /\  I  =  ( J  x.  K ) 
 /\  ( I  =  0  ->  J  <_  K ) )  /\  ( J  e.  NN0  /\  K  e.  NN0 ) )  ->  ( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) )
 
Theoremtrclrelexplem 36374* The union of relational powers to positive multiples of  N is a subset to the transitive closure raised to the power of  N. (Contributed by RP, 15-Jun-2020.)
 |-  ( N  e.  NN  ->  U_ k  e.  NN  (
 ( D ^r 
 k ) ^r  N )  C_  ( U_ j  e.  NN  ( D ^r  j ) ^r  N ) )
 
Theoremiunrelexpmin2 36375* The indexed union of relation exponentiation over the natural numbers (including zero) is the minimum reflexive-transitive relation that includes the relation. (Contributed by RP, 4-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  N  ( r ^r  n ) )   =>    |-  ( ( R  e.  V  /\  N  =  NN0 )  ->  A. s ( ( (  _I  |`  ( dom 
 R  u.  ran  R ) )  C_  s  /\  R  C_  s  /\  (
 s  o.  s ) 
 C_  s )  ->  ( C `  R ) 
 C_  s ) )
 
Theoremrelexp01min 36376 With exponents limited to 0 and 1, the composition of powers of a relation is the relation raised to the minimum of exponents. (Contributed by RP, 12-Jun-2020.)
 |-  (
 ( ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
 )  /\  ( J  e.  { 0 ,  1 }  /\  K  e.  { 0 ,  1 } ) )  ->  (
 ( R ^r  J ) ^r  K )  =  ( R ^r  I ) )
 
Theoremrelexp1idm 36377 Repeated raising a relation to the first power is idempotent. (Contributed by RP, 12-Jun-2020.)
 |-  ( R  e.  V  ->  ( ( R ^r 
 1 ) ^r 
 1 )  =  ( R ^r  1 ) )
 
Theoremrelexp0idm 36378 Repeated raising a relation to the zeroth power is idempotent. (Contributed by RP, 12-Jun-2020.)
 |-  ( R  e.  V  ->  ( ( R ^r 
 0 ) ^r 
 0 )  =  ( R ^r  0 ) )
 
Theoremrelexp0a 36379 Absorbtion law for zeroth power of a relation. (Contributed by RP, 17-Jun-2020.)
 |-  (
 ( A  e.  V  /\  N  e.  NN0 )  ->  ( ( A ^r  N ) ^r 
 0 )  C_  ( A ^r  0 ) )
 
Theoremrelexpxpmin 36380 The composition of powers of a cross-product of non-disjoint sets is the cross product raised to the minimum exponent. (Contributed by RP, 13-Jun-2020.)
 |-  (
 ( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) )  /\  ( I  =  if ( J  <  K ,  J ,  K )  /\  J  e.  NN0  /\  K  e.  NN0 ) )  ->  (
 ( ( A  X.  B ) ^r  J ) ^r  K )  =  (
 ( A  X.  B ) ^r  I ) )
 
Theoremrelexpaddss 36381 The composition of two powers of a relation is a subset of the relation raised to the sum of those exponents. This is equality where  R is a relation as shown by relexpaddd 13194 or when the sum of the powers isn't 1 as shown by relexpaddg 13193. (Contributed by RP, 3-Jun-2020.)
 |-  (
 ( N  e.  NN0  /\  M  e.  NN0  /\  R  e.  V )  ->  (
 ( R ^r  N )  o.  ( R ^r  M ) )  C_  ( R ^r  ( N  +  M ) ) )
 
Theoremiunrelexpuztr 36382* The indexed union of relation exponentiation over upper integers is a transive relation. Generalized from rtrclreclem3 13200. (Contributed by RP, 4-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  N  ( r ^r  n ) )   =>    |-  ( ( R  e.  V  /\  N  =  (
 ZZ>= `  M )  /\  M  e.  NN0 )  ->  ( ( C `  R )  o.  ( C `  R ) ) 
 C_  ( C `  R ) )
 
21.25.2.4  Transitive closure of a relation
 
Theoremdftrcl3 36383* Transitive closure of a relation, expressed as indexed union of powers of relations. (Contributed by RP, 5-Jun-2020.)
 |-  t+  =  ( r  e.  _V  |->  U_ n  e.  NN  ( r ^r  n ) )
 
Theorembrfvtrcld 36384* If two elements are connected by the transitive closure of a relation, then they are connected via 
n instances the relation, for some counting number  n. (Contributed by RP, 22-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( A ( t+ `
  R ) B  <->  E. n  e.  NN  A ( R ^r  n ) B ) )
 
Theoremfvtrcllb1d 36385 A set is a subset of its image under the transitive closure. (Contributed by RP, 22-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  R  C_  ( t+ `  R ) )
 
Theoremtrclfvcom 36386 The transitive closure of a relation commutes with the relation. (Contributed by RP, 18-Jul-2020.)
 |-  ( R  e.  V  ->  ( ( t+ `  R )  o.  R )  =  ( R  o.  ( t+ `  R ) ) )
 
Theoremcnvtrclfv 36387 The converse of the transitive closure is equal to the transitive closure of the converse relation. (Contributed by RP, 19-Jul-2020.)
 |-  ( R  e.  V  ->  `' ( t+ `  R )  =  (
 t+ `  `' R ) )
 
Theoremcotrcltrcl 36388 The transitive closure is idempotent. (Contributed by RP, 16-Jun-2020.)
 |-  (
 t+  o.  t+ )  =  t+
 
Theoremtrclimalb2 36389 Lower bound for image under a transitive closure. (Contributed by RP, 1-Jul-2020.)
 |-  (
 ( R  e.  V  /\  ( R " ( A  u.  B ) ) 
 C_  B )  ->  ( ( t+ `
  R ) " A )  C_  B )
 
Theorembrtrclfv2 36390* Two ways to indicate two elements are related by the transitive closure of a relation. (Contributed by RP, 1-Jul-2020.)
 |-  (
 ( X  e.  U  /\  Y  e.  V  /\  R  e.  W )  ->  ( X ( t+ `  R ) Y  <->  Y  e.  |^| { f  |  ( R " ( { X }  u.  f
 ) )  C_  f } ) )
 
Theoremtrclfvdecomr 36391 The transitive closure of a relation may be decomposed into a union of the relation and the composition of the relation with its transitive closure. (Contributed by RP, 18-Jul-2020.)
 |-  ( R  e.  V  ->  ( t+ `  R )  =  ( R  u.  ( ( t+ `
  R )  o.  R ) ) )
 
Theoremtrclfvdecoml 36392 The transitive closure of a relation may be decomposed into a union of the relation and the composition of the relation with its transitive closure. (Contributed by RP, 18-Jul-2020.)
 |-  ( R  e.  V  ->  ( t+ `  R )  =  ( R  u.  ( R  o.  (
 t+ `  R ) ) ) )
 
TheoremdmtrclfvRP 36393 The domain of the transitive closure is equal to the domain of the relation. (Contributed by RP, 18-Jul-2020.) (Proof modification is discouraged.)
 |-  ( R  e.  V  ->  dom  ( t+ `  R )  =  dom  R )
 
TheoremrntrclfvRP 36394 The range of the transitive closure is equal to the range of the relation. (Contributed by RP, 19-Jul-2020.) (Proof modification is discouraged.)
 |-  ( R  e.  V  ->  ran  ( t+ `  R )  =  ran  R )
 
Theoremrntrclfv 36395 The range of the transitive closure is equal to the range of the relation. (Contributed by RP, 18-Jul-2020.) (Proof modification is discouraged.)
 |-  ( R  e.  V  ->  ran  ( t+ `  R )  =  ran  R )
 
Theoremdfrtrcl3 36396* Reflexive-transitive closure of a relation, expressed as indexed union of powers of relations. Generalized from dfrtrcl2 13202. (Contributed by RP, 5-Jun-2020.)
 |-  t*  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^r  n ) )
 
Theorembrfvrtrcld 36397* If two elements are connected by the reflexive-transitive closure of a relation, then they are connected via  n instances the relation, for some natural number  n. Similar of dfrtrclrec2 13197. (Contributed by RP, 22-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( A ( t* `
  R ) B  <->  E. n  e.  NN0  A ( R ^r  n ) B ) )
 
Theoremfvrtrcllb0d 36398 A restriction of the identity relation is a subset of the reflexive-transitive closure of a set. (Contributed by RP, 22-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  (  _I  |`  ( dom  R  u.  ran  R ) ) 
 C_  ( t* `
  R ) )
 
Theoremfvrtrcllb0da 36399 A restriction of the identity relation is a subset of the reflexive-transitive closure of a relation. (Contributed by RP, 22-Jul-2020.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  (  _I  |`  U. U. R )  C_  ( t* `
  R ) )
 
Theoremfvrtrcllb1d 36400 A set is a subset of its image under the reflexive-transitive closure. (Contributed by RP, 22-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  R  C_  ( t* `  R ) )
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