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Theorem List for Metamath Proof Explorer - 36101-36200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremifpdfxor 36101 Define xor as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  \/_  ps )  <-> if- (
 ph ,  -.  ps ,  ps ) )
 
Theoremifpbi12 36102 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
 |-  (
 ( ( ph  <->  ps )  /\  ( ch 
 <-> 
 th ) )  ->  (if- ( ph ,  ch ,  ta )  <-> if- ( ps ,  th ,  ta ) ) )
 
Theoremifpbi13 36103 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
 |-  (
 ( ( ph  <->  ps )  /\  ( ch 
 <-> 
 th ) )  ->  (if- ( ph ,  ta ,  ch )  <-> if- ( ps ,  ta ,  th ) ) )
 
Theoremifpbi123 36104 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
 |-  (
 ( ( ph  <->  ps )  /\  ( ch 
 <-> 
 th )  /\  ( ta 
 <->  et ) )  ->  (if- ( ph ,  ch ,  ta )  <-> if- ( ps ,  th ,  et ) ) )
 
Theoremifpidg 36105 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( th  <-> if- ( ph ,  ps ,  ch ) )  <->  ( ( ( ( ph  /\  ps )  ->  th )  /\  (
 ( ph  /\  th )  ->  ps ) )  /\  ( ( ch  ->  (
 ph  \/  th )
 )  /\  ( th  ->  ( ph  \/  ch ) ) ) ) )
 
Theoremifpid3g 36106 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ch  <-> if- ( ph ,  ps ,  ch ) )  <->  ( ( (
 ph  /\  ps )  ->  ch )  /\  (
 ( ph  /\  ch )  ->  ps ) ) )
 
Theoremifpid2g 36107 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ps  <-> if- ( ph ,  ps ,  ch ) )  <->  ( ( ps 
 ->  ( ph  \/  ch ) )  /\  ( ch 
 ->  ( ph  \/  ps ) ) ) )
 
Theoremifpid1g 36108 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  <-> if- ( ph ,  ps ,  ch ) )  <->  ( ( ch 
 ->  ph )  /\  ( ph  ->  ps ) ) )
 
Theoremifpim23g 36109 Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
 |-  (
 ( ( ph  ->  ps )  <-> if- ( ch ,  ps ,  -.  ph ) )  <->  ( ( (
 ph  /\  ps )  ->  ch )  /\  ( ch  ->  ( ph  \/  ps ) ) ) )
 
Theoremifpim3 36110 Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
 |-  (
 ( ph  ->  ps )  <-> if- (
 ph ,  ps ,  -.  ph ) )
 
Theoremifpnim1 36111 Restate negated implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
 |-  ( -.  ( ph  ->  ps )  <-> if- (
 ph ,  -.  ps ,  ph ) )
 
Theoremifpim4 36112 Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
 |-  (
 ( ph  ->  ps )  <-> if- ( ps ,  ps ,  -.  ph ) )
 
Theoremifpnim2 36113 Restate negated implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
 |-  ( -.  ( ph  ->  ps )  <-> if- ( ps ,  -.  ps ,  ph ) )
 
Theoremifpim123g 36114 Implication of conditional logical operators. The right hand side is basically conjunctive normal form which is useful in proofs. (Contributed by RP, 16-Apr-2020.)
 |-  (
 (if- ( ph ,  ch ,  ta )  -> if- ( ps ,  th ,  et ) )  <->  ( ( ( ( ph  ->  -.  ps )  \/  ( ch  ->  th ) )  /\  (
 ( ps  ->  ph )  \/  ( ta  ->  th )
 ) )  /\  (
 ( ( ph  ->  ps )  \/  ( ch 
 ->  et ) )  /\  ( ( -.  ps  -> 
 ph )  \/  ( ta  ->  et ) ) ) ) )
 
Theoremifpim1g 36115 Implication of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
 |-  (
 (if- ( ph ,  ch ,  th )  -> if- ( ps ,  ch ,  th ) )  <->  ( ( ( ps  ->  ph )  \/  ( th  ->  ch )
 )  /\  ( ( ph  ->  ps )  \/  ( ch  ->  th ) ) ) )
 
Theoremifp1bi 36116 Substitute the first element of conditional logical operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 (if- ( ph ,  ch ,  th )  <-> if- ( ps ,  ch ,  th ) )  <->  ( ( ( ( ph  ->  ps )  \/  ( ch  ->  th )
 )  /\  ( ( ph  ->  ps )  \/  ( th  ->  ch ) ) ) 
 /\  ( ( ( ps  ->  ph )  \/  ( ch  ->  th )
 )  /\  ( ( ps  ->  ph )  \/  ( th  ->  ch ) ) ) ) )
 
Theoremifpbi1b 36117 When the first variable is irrelevant, it can be replaced. (Contributed by RP, 25-Apr-2020.)
 |-  (if- ( ph ,  ch ,  ch )  <-> if- ( ps ,  ch ,  ch ) )
 
Theoremifpimimb 36118 Factor conditional logic operator over implication in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
 |-  (if- ( ph ,  ( ps 
 ->  ch ) ,  ( th  ->  ta ) )  <->  (if- ( ph ,  ps ,  th )  -> if- ( ph ,  ch ,  ta ) ) )
 
Theoremifpororb 36119 Factor conditional logic operator over disjunction in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
 |-  (if- ( ph ,  ( ps 
 \/  ch ) ,  ( th  \/  ta ) )  <-> 
 (if- ( ph ,  ps ,  th )  \/ if-
 ( ph ,  ch ,  ta ) ) )
 
Theoremifpananb 36120 Factor conditional logic operator over conjunction in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
 |-  (if- ( ph ,  ( ps 
 /\  ch ) ,  ( th  /\  ta ) )  <-> 
 (if- ( ph ,  ps ,  th )  /\ if- (
 ph ,  ch ,  ta ) ) )
 
Theoremifpnannanb 36121 Factor conditional logic operator over nand in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
 |-  (if- ( ph ,  ( ps  -/\  ch ) ,  ( th  -/\  ta ) )  <-> 
 (if- ( ph ,  ps ,  th )  -/\ if- (
 ph ,  ch ,  ta ) ) )
 
Theoremifpor123g 36122 Disjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
 |-  (
 (if- ( ph ,  ch ,  ta )  \/ if-
 ( ps ,  th ,  et ) )  <->  ( ( ( ( ph  ->  -.  ps )  \/  ( ch  \/  th ) )  /\  (
 ( ps  ->  ph )  \/  ( ta  \/  th ) ) )  /\  ( ( ( ph  ->  ps )  \/  ( ch  \/  et ) ) 
 /\  ( ( -. 
 ps  ->  ph )  \/  ( ta  \/  et ) ) ) ) )
 
Theoremifpimim 36123 Consequnce of implication. (Contributed by RP, 17-Apr-2020.)
 |-  (if- ( ph ,  ( ps 
 ->  ch ) ,  ( th  ->  ta ) )  ->  (if- ( ph ,  ps ,  th )  -> if- ( ph ,  ch ,  ta )
 ) )
 
Theoremifpbibib 36124 Factor conditional logic operator over biimplication in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
 |-  (if- ( ph ,  ( ps  <->  ch ) ,  ( th  <->  ta ) )  <->  (if- ( ph ,  ps ,  th )  <-> if- ( ph ,  ch ,  ta ) ) )
 
Theoremifpxorxorb 36125 Factor conditional logic operator over xor in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
 |-  (if- ( ph ,  ( ps 
 \/_  ch ) ,  ( th  \/_  ta ) )  <-> 
 (if- ( ph ,  ps ,  th )  \/_ if- (
 ph ,  ch ,  ta ) ) )
 
21.25.1.2  Sophisms
 
Theoremrp-fakeimass 36126 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 36127 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 36128 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 36129 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 36130 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 36131 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 36132* 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 36133* 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 36134* 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 36135* 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 36136 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 36137 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 36138 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 7857 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 6593), various sets in the field of topology (inopn 19927, incld 20056, innei 20139, ... ) and "universal" classes like weak universes (wunin 9145, tskin 9191) and the class of all sets (inex1g 4567) .

 
Theoremfipjust 36139* 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 36140* 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 36141* 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 36142* 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 36143* 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 36144* 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 36145* The class of all subsets of a class is closed under set 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 36146* 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 36147* 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 36148* 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 36149 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 36150* 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 36151* 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 36152* 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 36153* 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 36154* 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 36155* 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 36156* 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 36157 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 36158* 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 36159* 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 36160 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 36161 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 36162 Subclass theorem for converse. (Contributed by RP, 22-Oct-2020.)
 |-  ( Rel  A  ->  ( A  C_  B  <->  `' A  C_  `' B ) )
 
Theoremrelnonrel 36163 The non-relation part of a relation is empty. (Contributed by RP, 22-Oct-2020.)
 |-  ( Rel  A  <->  ( A  \  `' `' A )  =  (/) )
 
Theoremcnvnonrel 36164 The converse of the non-relation part of a class is empty. (Contributed by RP, 18-Oct-2020.)
 |-  `' ( A  \  `' `' A )  =  (/)
 
Theorembrnonrel 36165 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 36166 The domain of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
 |-  dom  ( A  \  `' `' A )  =  (/)
 
Theoremrnnonrel 36167 The range of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
 |-  ran  ( A  \  `' `' A )  =  (/)
 
Theoremresnonrel 36168 A restriction of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
 |-  (
 ( A  \  `' `' A )  |`  B )  =  (/)
 
Theoremimanonrel 36169 An image under the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
 |-  (
 ( A  \  `' `' A ) " B )  =  (/)
 
Theoremcononrel1 36170 Composition with the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
 |-  (
 ( A  \  `' `' A )  o.  B )  =  (/)
 
Theoremcononrel2 36171 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 28229 by Thierry Arnoux.

 
Theoremelmapintab 36172* 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 36173 The function value of any class under a non-relation is empty. (Contributed by RP, 23-Oct-2020.)
 |-  (
 ( A  \  `' `' A ) `  X )  =  (/)
 
Theoremelinlem 36174 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 36175 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 36176* 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 36177 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 36178* 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 36179* 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 36180* 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 36181. (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 36181* 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 36182* 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 36183 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 36184 Equality implies bijection. (Contributed by RP, 24-Jul-2020.)
 |-  ( A  =  B  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A  =  B  ->  ( ( R 
 C_  A  /\  ph )  <->  ( R  C_  B  /\  ps ) ) )
 
Theoremcbvcllem 36185* 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 36186* 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 36187* 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 36188* 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 36189* Definition of identity relation as the trivial closure. (Contributed by RP, 26-Jul-2020.)
 |-  _I  =  ( x  e.  _V  |->  |^|
 { y  |  ( x  C_  y  /\ T.  ) } )
 
Theoremmptrcllem 36190* 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 36191 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 36192* 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 36193 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 36194* 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 36195* 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 36196* 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 36197* 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 36198* 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 36199* 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 36200* 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 ) } )
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