HomeHome Metamath Proof Explorer
Theorem List (p. 363 of 409)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-26620)
  Hilbert Space Explorer  Hilbert Space Explorer
(26621-28143)
  Users' Mathboxes  Users' Mathboxes
(28144-40813)
 

Theorem List for Metamath Proof Explorer - 36201-36300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
21.25.1.2  Sophisms
 
Theoremrp-fakeimass 36201 A special case where implication appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)
 |-  (
 ( ph  \/  ch )  <->  ( ( ( ph  ->  ps )  ->  ch )  <->  (
 ph  ->  ( ps  ->  ch ) ) ) )
 
Theoremrp-fakeanorass 36202 A special case where a mixture of and and or appears to conform to a mixed associative law. (Contributed by Richard Penner, 26-Feb-2020.)
 |-  (
 ( ch  ->  ph )  <->  ( ( ( ph  /\  ps )  \/  ch )  <->  ( ph  /\  ( ps  \/  ch ) ) ) )
 
Theoremrp-fakeoranass 36203 A special case where a mixture of or and and appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)
 |-  (
 ( ph  ->  ch )  <->  ( ( ( ph  \/  ps )  /\  ch )  <->  (
 ph  \/  ( ps  /\ 
 ch ) ) ) )
 
Theoremrp-fakenanass 36204 A special case where nand appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)
 |-  (
 ( ph  <->  ch )  <->  ( ( (
 ph  -/\  ps )  -/\  ch )  <->  ( ph  -/\  ( ps  -/\  ch ) ) ) )
 
Theoremrp-fakeinunass 36205 A special case where a mixture of intersection and union appears to conform to a mixed associative law. (Contributed by Richard Penner, 26-Feb-2020.)
 |-  ( C  C_  A  <->  ( ( A  i^i  B )  u.  C )  =  ( A  i^i  ( B  u.  C ) ) )
 
Theoremrp-fakeuninass 36206 A special case where a mixture of union and intersection appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)
 |-  ( A  C_  C  <->  ( ( A  u.  B )  i^i 
 C )  =  ( A  u.  ( B  i^i  C ) ) )
 
21.25.1.3  Finite Sets

Membership in the class of finite sets can be expressed in many ways.

 
Theoremrp-isfinite5 36207* A set is said to be finite if it can be put in one-to-one correspondence with all the natural numbers between 1 and some  n  e.  NN0. (Contributed by Richard Penner, 3-Mar-2020.)
 |-  ( A  e.  Fin  <->  E. n  e.  NN0  ( 1 ... n )  ~~  A )
 
Theoremrp-isfinite6 36208* A set is said to be finite if it is either empty or it can be put in one-to-one correspondence with all the natural numbers between 1 and some  n  e.  NN. (Contributed by Richard Penner, 10-Mar-2020.)
 |-  ( A  e.  Fin  <->  ( A  =  (/) 
 \/  E. n  e.  NN  ( 1 ... n )  ~~  A ) )
 
21.25.1.4  Infinite Sets
 
Theorempwelg 36209* The powerclass is an element of a class closed under union and powerclass operations iff the element is a member of that class. (Contributed by RP, 21-Mar-2020.)
 |-  ( A. x  e.  B  ( U. x  e.  B  /\  ~P x  e.  B )  ->  ( A  e.  B 
 <->  ~P A  e.  B ) )
 
Theorempwinfig 36210* The powerclass of an infinite set is an infinite set, and vice-versa. Here  B is a class which is closed under both the union and the powerclass operations and which may have infinite sets as members. (Contributed by RP, 21-Mar-2020.)
 |-  ( A. x  e.  B  ( U. x  e.  B  /\  ~P x  e.  B )  ->  ( A  e.  ( B  \  Fin )  <->  ~P A  e.  ( B 
 \  Fin ) ) )
 
Theorempwinfi2 36211 The powerclass of an infinite set is an infinite set, and vice-versa. Here  U is a weak universe. (Contributed by RP, 21-Mar-2020.)
 |-  ( U  e. WUni  ->  ( A  e.  ( U  \  Fin )  <->  ~P A  e.  ( U  \  Fin ) ) )
 
Theorempwinfi3 36212 The powerclass of an infinite set is an infinite set, and vice-versa. Here  T is a transitive Tarski universe. (Contributed by RP, 21-Mar-2020.)
 |-  (
 ( T  e.  Tarski  /\ 
 Tr  T )  ->  ( A  e.  ( T  \  Fin )  <->  ~P A  e.  ( T  \  Fin ) ) )
 
Theorempwinfi 36213 The powerclass of an infinite set is an infinite set, and vice-versa. (Contributed by RP, 21-Mar-2020.)
 |-  ( A  e.  ( _V  \ 
 Fin )  <->  ~P A  e.  ( _V  \  Fin ) )
 
21.25.1.5  Finite intersection property

While there is not yet a definition, the finite intersection property of a class is introduced by fiint 7874 where two textbook definitions are shown to be equivalent.

This property is seen often with ordinal numbers (onin 5473, ordelinel 5540 ), chains of sets ordered by the proper subset relation (sorpssin 6606), various sets in the field of topology (inopn 19978, incld 20107, innei 20190, ... ) and "universal" classes like weak universes (wunin 9164, tskin 9210) and the class of all sets (inex1g 4560) .

 
Theoremfipjust 36214* A definition of the finite intersection property of a class based on closure under pair-wise intersection of its elements is independent of the dummy variables. (Contributed by Richard Penner, 1-Jan-2020.)
 |-  ( A. u  e.  A  A. v  e.  A  ( u  i^i  v )  e.  A  <->  A. x  e.  A  A. y  e.  A  ( x  i^i  y )  e.  A )
 
Theoremcllem0 36215* The class of all sets with property  ph ( z ) is closed under the binary operation on sets defined in  R ( x ,  y ). (Contributed by Richard Penner, 3-Jan-2020.)
 |-  V  =  { z  |  ph }   &    |-  R  e.  U   &    |-  ( z  =  R  ->  ( ph  <->  ps ) )   &    |-  ( z  =  x  ->  ( ph  <->  ch ) )   &    |-  ( z  =  y  ->  ( ph  <->  th ) )   &    |-  ( ( ch 
 /\  th )  ->  ps )   =>    |-  A. x  e.  V  A. y  e.  V  R  e.  V
 
Theoremsuperficl 36216* The class of all supersets of a class has the finite intersection property. (Contributed by Richard Penner, 1-Jan-2020.) (Proof shortened by Richard Penner, 3-Jan-2020.)
 |-  A  =  { z  |  B  C_  z }   =>    |- 
 A. x  e.  A  A. y  e.  A  ( x  i^i  y )  e.  A
 
Theoremsuperuncl 36217* The class of all supersets of a class is closed under binary union. (Contributed by Richard Penner, 3-Jan-2020.)
 |-  A  =  { z  |  B  C_  z }   =>    |- 
 A. x  e.  A  A. y  e.  A  ( x  u.  y )  e.  A
 
Theoremssficl 36218* The class of all subsets of a class has the finite intersection property. (Contributed by Richard Penner, 1-Jan-2020.) (Proof shortened by Richard Penner, 3-Jan-2020.)
 |-  A  =  { z  |  z 
 C_  B }   =>    |-  A. x  e.  A  A. y  e.  A  ( x  i^i  y )  e.  A
 
Theoremssuncl 36219* The class of all subsets of a class is closed under binary union. (Contributed by Richard Penner, 3-Jan-2020.)
 |-  A  =  { z  |  z 
 C_  B }   =>    |-  A. x  e.  A  A. y  e.  A  ( x  u.  y )  e.  A
 
Theoremssdifcl 36220* The class of all subsets of a class is closed under class difference. (Contributed by Richard Penner, 3-Jan-2020.)
 |-  A  =  { z  |  z 
 C_  B }   =>    |-  A. x  e.  A  A. y  e.  A  ( x  \  y )  e.  A
 
Theoremsssymdifcl 36221* The class of all subsets of a class is closed under symmetric difference. (Contributed by Richard Penner, 3-Jan-2020.)
 |-  A  =  { z  |  z 
 C_  B }   =>    |-  A. x  e.  A  A. y  e.  A  ( ( x 
 \  y )  u.  ( y  \  x ) )  e.  A
 
Theoremfiinfi 36222* If two classes have the finite intersection property, then so does their intersection. (Contributed by Richard Penner, 1-Jan-2020.)
 |-  ( ph  ->  A. x  e.  A  A. y  e.  A  ( x  i^i  y )  e.  A )   &    |-  ( ph  ->  A. x  e.  B  A. y  e.  B  ( x  i^i  y )  e.  B )   &    |-  ( ph  ->  C  =  ( A  i^i  B ) )   =>    |-  ( ph  ->  A. x  e.  C  A. y  e.  C  ( x  i^i  y )  e.  C )
 
21.25.1.6  RP ADDTO: The universal class
 
Theoremelabd 36223* Explicit demonstration the class 
{ x  |  ps } is not empty by the example  X. (Contributed by RP, 12-Aug-2020.)
 |-  ( ph  ->  X  e.  _V )   &    |-  ( ph  ->  ch )   &    |-  ( x  =  X  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  E. x ps )
 
21.25.1.7  RP ADDTO: Subclasses and subsets
 
Theoremrababg 36224 Condition when restricted class is equal to unrestricted class. (Contributed by RP, 13-Aug-2020.)
 |-  ( A. x ( ph  ->  x  e.  A )  <->  { x  e.  A  |  ph }  =  { x  |  ph } )
 
21.25.1.8  RP ADDTO: The intersection of a class
 
Theoremelintabg 36225* Two ways of saying a set is an element of the intersection of a class. (Contributed by RP, 13-Aug-2020.)
 |-  ( A  e.  V  ->  ( A  e.  |^| { x  |  ph }  <->  A. x ( ph  ->  A  e.  x ) ) )
 
Theoremelinintab 36226* Two ways of saying a set is an element of the intersection of a class with the intersection of a class. (Contributed by RP, 13-Aug-2020.)
 |-  ( A  e.  ( B  i^i  |^| { x  |  ph
 } )  <->  ( A  e.  B  /\  A. x (
 ph  ->  A  e.  x ) ) )
 
Theoremelmapintrab 36227* Two ways to say a set is an element of the intersection of a class of images. (Contributed by RP, 16-Aug-2020.)
 |-  C  e.  _V   &    |-  C  C_  B   =>    |-  ( A  e.  V  ->  ( A  e.  |^| { w  e.  ~P B  |  E. x ( w  =  C  /\  ph ) } 
 <->  ( ( E. x ph 
 ->  A  e.  B ) 
 /\  A. x ( ph  ->  A  e.  C ) ) ) )
 
21.25.1.9  RP ADDTO: Theorems requiring subset and intersection existence
 
Theoremelinintrab 36228* Two ways of saying a set is an element of the intersection of a class with the intersection of a class. (Contributed by RP, 14-Aug-2020.)
 |-  ( A  e.  V  ->  ( A  e.  |^| { w  e.  ~P B  |  E. x ( w  =  ( B  i^i  x )  /\  ph ) }  <->  ( ( E. x ph  ->  A  e.  B )  /\  A. x ( ph  ->  A  e.  x ) ) ) )
 
Theoreminintabss 36229* Upper bound on intersection of class and the intersection of a class. (Contributed by RP, 13-Aug-2020.)
 |-  ( A  i^i  |^| { x  |  ph
 } )  C_  |^| { w  e.  ~P A  |  E. x ( w  =  ( A  i^i  x )  /\  ph ) }
 
Theoreminintabd 36230* Value of the intersection of class with the intersection of a non-empty class. (Contributed by RP, 13-Aug-2020.)
 |-  ( ph  ->  E. x ps )   =>    |-  ( ph  ->  ( A  i^i  |^|
 { x  |  ps } )  =  |^| { w  e.  ~P A  |  E. x ( w  =  ( A  i^i  x )  /\  ps ) }
 )
 
21.25.1.10  RP ADDTO: Relations
 
Theoremxpinintabd 36231* Value of the intersection of cross-product with the intersection of a non-empty class. (Contributed by RP, 12-Aug-2020.)
 |-  ( ph  ->  E. x ps )   =>    |-  ( ph  ->  ( ( A  X.  B )  i^i  |^| { x  |  ps } )  =  |^| { w  e.  ~P ( A  X.  B )  |  E. x ( w  =  (
 ( A  X.  B )  i^i  x )  /\  ps ) } )
 
Theoremrelintabex 36232 If the intersection of a class is a relation, then the class is non-empty. (Contributed by RP, 12-Aug-2020.)
 |-  ( Rel  |^| { x  |  ph
 }  ->  E. x ph )
 
Theoremelcnvcnvintab 36233* Two ways of saying a set is an element of the converse of the converse of the intersection of a class. (Contributed by RP, 20-Aug-2020.)
 |-  ( A  e.  `' `' |^|
 { x  |  ph }  <-> 
 ( A  e.  ( _V  X.  _V )  /\  A. x ( ph  ->  A  e.  x ) ) )
 
Theoremrelintab 36234* Value of the intersection of a class when it is a relation. (Contributed by RP, 12-Aug-2020.)
 |-  ( Rel  |^| { x  |  ph
 }  ->  |^| { x  |  ph }  =  |^| { w  e.  ~P ( _V  X.  _V )  | 
 E. x ( w  =  `' `' x  /\  ph ) } )
 
Theoremnonrel 36235 A non-relation is equal to the base class with all ordered pairs removed. (Contributed by RP, 25-Oct-2020.)
 |-  ( A  \  `' `' A )  =  ( A  \  ( _V  X.  _V ) )
 
Theoremelnonrel 36236 Only an ordered pair where not both entries are sets could be an element of the non-relation part of class. (Contributed by RP, 25-Oct-2020.)
 |-  ( <. X ,  Y >.  e.  ( A  \  `' `' A )  <->  ( (/)  e.  A  /\  -.  ( X  e.  _V 
 /\  Y  e.  _V ) ) )
 
Theoremcnvssb 36237 Subclass theorem for converse. (Contributed by RP, 22-Oct-2020.)
 |-  ( Rel  A  ->  ( A  C_  B  <->  `' A  C_  `' B ) )
 
Theoremrelnonrel 36238 The non-relation part of a relation is empty. (Contributed by RP, 22-Oct-2020.)
 |-  ( Rel  A  <->  ( A  \  `' `' A )  =  (/) )
 
Theoremcnvnonrel 36239 The converse of the non-relation part of a class is empty. (Contributed by RP, 18-Oct-2020.)
 |-  `' ( A  \  `' `' A )  =  (/)
 
Theorembrnonrel 36240 A non-relation cannot relate any two classes. (Contributed by RP, 23-Oct-2020.)
 |-  (
 ( X  e.  U  /\  Y  e.  V ) 
 ->  -.  X ( A 
 \  `' `' A ) Y )
 
Theoremdmnonrel 36241 The domain of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
 |-  dom  ( A  \  `' `' A )  =  (/)
 
Theoremrnnonrel 36242 The range of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
 |-  ran  ( A  \  `' `' A )  =  (/)
 
Theoremresnonrel 36243 A restriction of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
 |-  (
 ( A  \  `' `' A )  |`  B )  =  (/)
 
Theoremimanonrel 36244 An image under the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
 |-  (
 ( A  \  `' `' A ) " B )  =  (/)
 
Theoremcononrel1 36245 Composition with the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
 |-  (
 ( A  \  `' `' A )  o.  B )  =  (/)
 
Theoremcononrel2 36246 Composition with the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
 |-  ( A  o.  ( B  \  `' `' B ) )  =  (/)
 
21.25.1.11  RP ADDTO: Functions

See also idssxp 28276 by Thierry Arnoux.

 
Theoremelmapintab 36247* Two ways to say a set is an element of mapped intersection of a class. Here  F maps elements of  C to elements of  |^| { x  | 
ph } or  x. (Contributed by RP, 19-Aug-2020.)
 |-  ( A  e.  B  <->  ( A  e.  C  /\  ( F `  A )  e.  |^| { x  |  ph } ) )   &    |-  ( A  e.  E  <->  ( A  e.  C  /\  ( F `  A )  e.  x ) )   =>    |-  ( A  e.  B  <->  ( A  e.  C  /\  A. x ( ph  ->  A  e.  E ) ) )
 
Theoremfvnonrel 36248 The function value of any class under a non-relation is empty. (Contributed by RP, 23-Oct-2020.)
 |-  (
 ( A  \  `' `' A ) `  X )  =  (/)
 
Theoremelinlem 36249 Two ways to say a set is a member of an intersection. (Contributed by RP, 19-Aug-2020.)
 |-  ( A  e.  ( B  i^i  C )  <->  ( A  e.  B  /\  (  _I  `  A )  e.  C ) )
 
Theoremelcnvcnvlem 36250 Two ways to say a set is a member of the converse of the converse of a class. (Contributed by RP, 20-Aug-2020.)
 |-  ( A  e.  `' `' B 
 <->  ( A  e.  ( _V  X.  _V )  /\  (  _I  `  A )  e.  B ) )
 
21.25.1.12  RP ADDTO: Finite induction (for finite ordinals)

Original probably needs new subsection for Relation-related existence theorems.

 
Theoremcnvcnvintabd 36251* Value of the relationship content of the intersection of a class. (Contributed by RP, 20-Aug-2020.)
 |-  ( ph  ->  E. x ps )   =>    |-  ( ph  ->  `' `' |^| { x  |  ps }  =  |^| { w  e. 
 ~P ( _V  X.  _V )  |  E. x ( w  =  `' `' x  /\  ps ) } )
 
21.25.1.13  RP ADDTO: First and second members of an ordered pair
 
Theoremelcnvlem 36252 Two ways to say a set is a member of the converse of a class. (Contributed by RP, 19-Aug-2020.)
 |-  F  =  ( x  e.  ( _V  X.  _V )  |->  <.
 ( 2nd `  x ) ,  ( 1st `  x ) >. )   =>    |-  ( A  e.  `' B 
 <->  ( A  e.  ( _V  X.  _V )  /\  ( F `  A )  e.  B ) )
 
Theoremelcnvintab 36253* Two ways of saying a set is an element of the converse of the intersection of a class. (Contributed by RP, 19-Aug-2020.)
 |-  ( A  e.  `' |^| { x  |  ph }  <->  ( A  e.  ( _V  X.  _V )  /\  A. x ( ph  ->  A  e.  `' x ) ) )
 
Theoremcnvintabd 36254* Value of the converse of the intersection of a non-empty class. (Contributed by RP, 20-Aug-2020.)
 |-  ( ph  ->  E. x ps )   =>    |-  ( ph  ->  `' |^| { x  |  ps }  =  |^| { w  e.  ~P ( _V  X.  _V )  | 
 E. x ( w  =  `' x  /\  ps ) } )
 
21.25.1.14  RP ADDTO: The reflexive and transitive properties of relations
 
Theoremundmrnresiss 36255* Two ways of saying the identity relation restricted to the union of the domain and range of a relation is a subset of a relation. Generalization of reflexg 36256. (Contributed by RP, 26-Sep-2020.)
 |-  (
 (  _I  |`  ( dom 
 A  u.  ran  A ) )  C_  B  <->  A. x A. y
 ( x A y 
 ->  ( x B x 
 /\  y B y ) ) )
 
Theoremreflexg 36256* Two ways of saying a relation is reflexive over its domain and range. (Contributed by RP, 4-Aug-2020.)
 |-  (
 (  _I  |`  ( dom 
 A  u.  ran  A ) )  C_  A  <->  A. x A. y
 ( x A y 
 ->  ( x A x 
 /\  y A y ) ) )
 
Theoremcnvssco 36257* A condition weaker than reflexivity. (Contributed by RP, 3-Aug-2020.)
 |-  ( `' A  C_  `' ( B  o.  C )  <->  A. x A. y E. z ( x A y  ->  ( x C z  /\  z B y ) ) )
 
Theoremrefimssco 36258 Reflexive relations are subsets of their self-composition. (Contributed by RP, 4-Aug-2020.)
 |-  (
 (  _I  |`  ( dom 
 A  u.  ran  A ) )  C_  A  ->  `' A  C_  `' ( A  o.  A ) )
 
21.25.1.15  RP ADDTO: Basic properties of closures
 
Theoremcleq2lem 36259 Equality implies bijection. (Contributed by RP, 24-Jul-2020.)
 |-  ( A  =  B  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A  =  B  ->  ( ( R 
 C_  A  /\  ph )  <->  ( R  C_  B  /\  ps ) ) )
 
Theoremcbvcllem 36260* Change of bound variable in class of supersets of a with a property. (Contributed by RP, 24-Jul-2020.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  { x  |  ( X  C_  x  /\  ph ) }  =  {
 y  |  ( X 
 C_  y  /\  ps ) }
 
Theoremintabssd 36261* When for each element  y there is a subset  A which may substituted for  x such that  y satisfying  ch implies  x satisfies  ps then the intersection of all  x that satisfy  ps is a subclass the intersection of all  y that satisfy  ch. (Contributed by RP, 17-Oct-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  =  A )  ->  ( ch  ->  ps ) )   &    |-  ( ph  ->  A  C_  y
 )   =>    |-  ( ph  ->  |^| { x  |  ps }  C_  |^| { y  |  ch } )
 
Theoremclublem 36262* If a superset  Y of  X possesses the property parameterized in  x in  ps, then  Y is a superset of the closure of that property for the set  X. (Contributed by RP, 23-Jul-2020.)
 |-  ( ph  ->  Y  e.  _V )   &    |-  ( x  =  Y  ->  ( ps  <->  ch ) )   &    |-  ( ph  ->  X  C_  Y )   &    |-  ( ph  ->  ch )   =>    |-  ( ph  ->  |^| { x  |  ( X  C_  x  /\  ps ) }  C_  Y )
 
Theoremclss2lem 36263* The closure of a property is a superset of the closure of a less restrictive property. (Contributed by RP, 24-Jul-2020.)
 |-  ( ph  ->  ( ch  ->  ps ) )   =>    |-  ( ph  ->  |^| { x  |  ( X  C_  x  /\  ps ) }  C_  |^|
 { x  |  ( X  C_  x  /\  ch ) } )
 
Theoremdfid7 36264* Definition of identity relation as the trivial closure. (Contributed by RP, 26-Jul-2020.)
 |-  _I  =  ( x  e.  _V  |->  |^|
 { y  |  ( x  C_  y  /\ T.  ) } )
 
Theoremmptrcllem 36265* Show two versions of a closure with reflexive properties are equal. (Contributed by RP, 19-Oct-2020.)
 |-  ( x  e.  V  ->  |^|
 { y  |  ( x  C_  y  /\  ( ph  /\  (  _I  |`  ( dom  y  u. 
 ran  y ) ) 
 C_  y ) ) }  e.  _V )   &    |-  ( x  e.  V  ->  |^|
 { z  |  ( ( x  u.  (  _I  |`  ( dom  x  u.  ran  x ) ) )  C_  z  /\  ps ) }  e.  _V )   &    |-  ( x  e.  V  ->  ch )   &    |-  ( x  e.  V  ->  th )   &    |-  ( x  e.  V  ->  ta )   &    |-  ( y  = 
 |^| { z  |  ( ( x  u.  (  _I  |`  ( dom  x  u.  ran  x ) ) )  C_  z  /\  ps ) }  ->  ( ph 
 <->  ch ) )   &    |-  (
 y  =  |^| { z  |  ( ( x  u.  (  _I  |`  ( dom  x  u.  ran  x )
 ) )  C_  z  /\  ps ) }  ->  ( (  _I  |`  ( dom  y  u.  ran  y
 ) )  C_  y  <->  th ) )   &    |-  ( z  = 
 |^| { y  |  ( x  C_  y  /\  ( ph  /\  (  _I  |`  ( dom  y  u. 
 ran  y ) ) 
 C_  y ) ) }  ->  ( ps  <->  ta ) )   =>    |-  ( x  e.  V  |->  |^|
 { y  |  ( x  C_  y  /\  ( ph  /\  (  _I  |`  ( dom  y  u. 
 ran  y ) ) 
 C_  y ) ) } )  =  ( x  e.  V  |->  |^| { z  |  ( ( x  u.  (  _I  |`  ( dom  x  u.  ran 
 x ) ) ) 
 C_  z  /\  ps ) } )
 
Theoremcotrintab 36266 The intersection of a class is a transitive relation if membership in the class implies the member is a transitive relation. (Contributed by RP, 28-Oct-2020.)
 |-  ( ph  ->  ( x  o.  x )  C_  x )   =>    |-  ( |^| { x  |  ph
 }  o.  |^| { x  |  ph } )  C_  |^|
 { x  |  ph }
 
Theoremrclexi 36267* The reflexive closure of a set exists. (Contributed by RP, 27-Oct-2020.)
 |-  A  e.  V   =>    |- 
 |^| { x  |  ( A  C_  x  /\  (  _I  |`  ( dom  x  u.  ran  x )
 )  C_  x ) }  e.  _V
 
Theoremrtrclexlem 36268 Existence of relation implies existence of union with Cartesian product of domain and range. (Contributed by RP, 1-Nov-2020.)
 |-  ( R  e.  V  ->  ( R  u.  ( ( dom  R  u.  ran  R )  X.  ( dom 
 R  u.  ran  R ) ) )  e. 
 _V )
 
Theoremrtrclex 36269* The reflexive-transitive closure of a set exists. (Contributed by RP, 1-Nov-2020.)
 |-  ( A  e.  _V  <->  |^| { x  |  ( A  C_  x  /\  ( ( x  o.  x )  C_  x  /\  (  _I  |`  ( dom  x  u.  ran  x )
 )  C_  x )
 ) }  e.  _V )
 
TheoremtrclubgNEW 36270* If a relation exists then the transitive closure has an upper bound. (Contributed by RP, 24-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  |^| { x  |  ( R  C_  x  /\  ( x  o.  x )  C_  x ) }  C_  ( R  u.  ( dom  R  X.  ran  R ) ) )
 
TheoremtrclubNEW 36271* If a relation exists then the transitive closure has an upper bound. (Contributed by RP, 24-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   &    |-  ( ph  ->  Rel  R )   =>    |-  ( ph  ->  |^| { x  |  ( R  C_  x  /\  ( x  o.  x )  C_  x ) }  C_  ( dom  R  X.  ran  R ) )
 
Theoremtrclexi 36272* The transitive closure of a set exists. (Contributed by RP, 27-Oct-2020.)
 |-  A  e.  V   =>    |- 
 |^| { x  |  ( A  C_  x  /\  ( x  o.  x )  C_  x ) }  e.  _V
 
Theoremrtrclexi 36273* The reflexive-transitive closure of a set exists. (Contributed by RP, 27-Oct-2020.)
 |-  A  e.  V   =>    |- 
 |^| { x  |  ( A  C_  x  /\  ( ( x  o.  x )  C_  x  /\  (  _I  |`  ( dom  x  u.  ran  x )
 )  C_  x )
 ) }  e.  _V
 
Theoremclrellem 36274* When the property  ps holds for a relation substituted for 
x, then the closure on that property is a relation if the base set is a relation. (Contributed by RP, 30-Jul-2020.)
 |-  ( ph  ->  Y  e.  _V )   &    |-  ( ph  ->  Rel  X )   &    |-  ( x  =  `' `' Y  ->  ( ps  <->  ch ) )   &    |-  ( ph  ->  X 
 C_  Y )   &    |-  ( ph  ->  ch )   =>    |-  ( ph  ->  Rel  |^| { x  |  ( X  C_  x  /\  ps ) } )
 
Theoremclcnvlem 36275* 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 36276* 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 36277* 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 36278* 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 36279* 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 36280* 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 36281* 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 36282 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 36283 Version of ax-4 1693 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 36284* 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 36285* 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 36286 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 36287* Converse of indexed union. (Contributed by RP, 20-Jun-2020.)
 |-  `' U_ x  e.  A  B  =  U_ x  e.  A  `' B
 
Theoremimaiun1 36288* 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 36289* Composition with an indexed union. Proof analgous to that of coiun 5364. (Contributed by RP, 20-Jun-2020.)
 |-  ( U_ x  e.  C  A  o.  B )  = 
 U_ x  e.  C  ( A  o.  B )
 
Theoremelintima 36290* 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 36291* 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 36292* 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 36293* 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 36294* 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 36295* 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 36296* 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 36297* 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 36298* 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 36299* 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 36300 Deduction about composition with a class with no relational content. (Contributed by Richard Penner, 24-Dec-2019.)
 |-  ( ph  ->  `' A  =  (/) )   =>    |-  ( ph  ->  ( A  o.  B )  =  (/) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40813
  Copyright terms: Public domain < Previous  Next >