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Theorem List for Metamath Proof Explorer - 37801-37900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembj-ifbi2 37801 Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
 |-  (
 ( ph  <->  ps )  ->  (if- ( ch ,  ph ,  th )  <-> if- ( ch ,  ps ,  th ) ) )
 
Theorembj-ifbi3 37802 Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
 |-  (
 ( ph  <->  ps )  ->  (if- ( ch ,  th ,  ph )  <-> if- ( ch ,  th ,  ps ) ) )
 
Theorembj-ifbi12 37803 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
 |-  (
 ( ( ph  <->  ps )  /\  ( ch 
 <-> 
 th ) )  ->  (if- ( ph ,  ch ,  ta )  <-> if- ( ps ,  th ,  ta ) ) )
 
Theorembj-ifbi13 37804 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
 |-  (
 ( ( ph  <->  ps )  /\  ( ch 
 <-> 
 th ) )  ->  (if- ( ph ,  ta ,  ch )  <-> if- ( ps ,  ta ,  th ) ) )
 
Theorembj-ifbi23 37805 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
 |-  (
 ( ( ph  <->  ps )  /\  ( ch 
 <-> 
 th ) )  ->  (if- ( ta ,  ph ,  ch )  <-> if- ( ta ,  ps ,  th ) ) )
 
Theorembj-ifbi123 37806 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 ) ) )
 
Theorembj-ifim123g 37807 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 ) ) ) ) )
 
Theorembj-ifidg 37808 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 ) ) ) ) )
 
Theorembj-ifid1g 37809 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  <-> if- ( ph ,  ps ,  ch ) )  <->  ( ( ch 
 ->  ph )  /\  ( ph  ->  ps ) ) )
 
Theorembj-ifid2g 37810 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ps  <-> if- ( ph ,  ps ,  ch ) )  <->  ( ( ps 
 ->  ( ph  \/  ch ) )  /\  ( ch 
 ->  ( ph  \/  ps ) ) ) )
 
Theorembj-ifid3g 37811 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ch  <-> if- ( ph ,  ps ,  ch ) )  <->  ( ( (
 ph  /\  ps )  ->  ch )  /\  (
 ( ph  /\  ch )  ->  ps ) ) )
 
Theorembj-ifid2 37812 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  ( ph 
 <-> if- ( ph , T.  , F.  ) )
 
Theorembj-ifim1 37813 Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  ->  ps )  <-> if- ( -.  ph , T.  ,  ps ) )
 
Theorembj-ifim2 37814 Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  ->  ps )  <-> if- ( ps , T.  ,  -.  ph ) )
 
Theorembj-ifim1g 37815 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 ) ) ) )
 
Theorembj-ifimimb 37816 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 ) ) )
 
Theorembj-ifimim 37817 Consequnce of implication. (Contributed by RP, 17-Apr-2020.)
 |-  (if- ( ph ,  ( ps 
 ->  ch ) ,  ( th  ->  ta ) )  ->  (if- ( ph ,  ps ,  th )  -> if- ( ph ,  ch ,  ta )
 ) )
 
Theorembj-ifnot 37818 Restate negated wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  ( -.  ph  <-> if- ( ph , F.  , T.  ) )
 
Theorembj-ifnot23 37819 Negation of conditional logical operator. (Contributed by RP, 18-Apr-2020.)
 |-  ( -. if- ( ph ,  ps ,  ch )  <-> if- ( ph ,  -.  ps ,  -.  ch )
 )
 
Theorembj-ifnotnotb 37820 Factor conditional logic operator over negation in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
 |-  (if- ( ph ,  -.  ps ,  -.  ch )  <->  -. if- ( ph ,  ps ,  ch ) )
 
Theorembj-if1bi 37821 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 ) ) ) ) )
 
Theorembj-ifbibib 37822 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 ) ) )
 
Theorembj-ifdfor 37823 Define or in terms of conditional logic operator and true. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  \/  ps )  <-> if- (
 ph , T.  ,  ps ) )
 
Theorembj-ifdfor2 37824 Define or in terms of conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  \/  ps )  <-> if- (
 ph ,  ph ,  ps ) )
 
Theorembj-iforcor 37825 Corollary of communtation of or. (Contributed by RP, 20-Apr-2020.)
 |-  (if- ( ph ,  ph ,  ps )  <-> if- ( ps ,  ps ,  ph ) )
 
Theorembj-ifor123g 37826 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 ) ) ) ) )
 
Theorembj-ifororb 37827 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 ) ) )
 
Theorembj-ifdfan 37828 Define and with conditional logic operator and false. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  /\  ps )  <-> if- (
 ph ,  ps , F.  ) )
 
Theorembj-ifan123g 37829 Conjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
 |-  (
 (if- ( ph ,  ch ,  ta )  /\ if- ( ps ,  th ,  et ) )  <->  ( ( ( -.  ph  \/  ch )  /\  ( ph  \/  ta ) )  /\  ( ( -.  ps  \/  th )  /\  ( ps  \/  et ) ) ) )
 
Theorembj-ifan23 37830 Conjunction of conditional logical operators. (Contributed by RP, 20-Apr-2020.)
 |-  (
 (if- ( ph ,  ps ,  ch )  /\ if- (
 ph ,  th ,  ta ) )  <-> if- ( ph ,  ( ps  /\  th ) ,  ( ch  /\  ta ) ) )
 
Theorembj-ifdfbi 37831 Define biimplication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  <->  ps )  <-> if- ( ph ,  ps ,  -.  ps ) )
 
Theorembj-ifananb 37832 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 ) ) )
 
Theorembj-ifdfxor 37833 Define xor as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  \/_  ps )  <-> if- (
 ph ,  -.  ps ,  ps ) )
 
Theorembj-ifxorxorb 37834 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 ) ) )
 
Theorembj-ifdfnan 37835 Define nand as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
 |-  (
 ( ph  -/\  ps )  <-> if- (
 ph ,  -.  ps , T.  ) )
 
Theorembj-ifnannanb 37836 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 ) ) )
 
21.32.1.2  Sophisms
 
Theoremrp-fakeimass 37837 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 37838 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 37839 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 37840 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 37841 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 37842 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 ) ) )
 
21.32.1.3  Finite Sets

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

 
Theoremrp-isfinite4 37843 A finite set is equinumerous to the range of integers from one up to the hash value of the set. In other words, counting objects with natural numbers works if and only if it is a finite collection. (Contributed by Richard Penner, 26-Feb-2020.)
 |-  ( A  e.  Fin  <->  ( 1 ... ( # `  A ) )  ~~  A )
 
Theoremrp-isfinite5 37844* 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 37845* 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 ) )
 
21.32.1.4  Infinite Sets
 
Theorempwelg 37846* 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 37847* 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 37848 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 37849 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 37850 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 ) )
 
21.32.1.5  Finite intersection property

While there is not yet a definition, the finite intersection property of a class is introduced by fiint 7815 where two textbook definitions are shown to be equivalent.

This property is seen often with ordinal numbers (onin 4918, ordelinel 4985 ), chains of sets ordered by the proper subset relation (sorpssin 6587), various sets in the field of topology (inopn 19534, incld 19670, innei 19752, ... ) and "universal" classes like weak universes (wunin 9108, tskin 9154) and the class of all sets (inex1g 4599) .

 
Theoremfipjust 37851* 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 37852* 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 37853* 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 37854* 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 37855* 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 37856* 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 37857* 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 37858* 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 37859* 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 )
 
21.32.2  Additional statements on relations and subclasses
 
Theoremcoss12d 37860 Subset deduction for composition of two classes. (Contributed by Richard Penner, 24-Dec-2019.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  C  C_  D )   =>    |-  ( ph  ->  ( A  o.  C )  C_  ( B  o.  D ) )
 
Theoremcoemptyd 37861 Deduction about composition of classes with no relational content in common. (Contributed by Richard Penner, 24-Dec-2019.)
 |-  ( ph  ->  ( dom  A  i^i  ran  B )  =  (/) )   =>    |-  ( ph  ->  ( A  o.  B )  =  (/) )
 
Theoremconrel1d 37862 Deduction about composition with a class with no relational content. (Contributed by Richard Penner, 24-Dec-2019.)
 |-  ( ph  ->  `' A  =  (/) )   =>    |-  ( ph  ->  ( A  o.  B )  =  (/) )
 
Theoremconrel2d 37863 Deduction about composition with a class with no relational content. (Contributed by Richard Penner, 24-Dec-2019.)
 |-  ( ph  ->  `' A  =  (/) )   =>    |-  ( ph  ->  ( B  o.  A )  =  (/) )
 
Theorembrintclab 37864* Two ways to express a binary relation which is the intersection of a class. (Contributed by RP, 4-Apr-2020.)
 |-  ( A |^| { x  |  ph
 } B  <->  A. x ( ph  -> 
 <. A ,  B >.  e.  x ) )
 
Theoremelimaint 37865* 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
 )
 
Theoremelintima 37866* 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 )
 
Theoremal3im 37867 Version of ax-4 1632 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 37868* 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 ) }
 
Theoremintimass 37869* 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 37870* 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 37871* 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 37872* 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 37873* 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 }
 ) )
 
21.32.2.1  Transitive relations (not to be confused with transitive classes).
 
Theoremxpcogend 37874 The most interesting case of the composition of two cross products. (Contributed by Richard Penner, 24-Dec-2019.)
 |-  ( ph  ->  ( B  i^i  C )  =/=  (/) )   =>    |-  ( ph  ->  ( ( C  X.  D )  o.  ( A  X.  B ) )  =  ( A  X.  D ) )
 
Theoremxpcoidgend 37875 If two classes are not disjoint, then the composition of their cross-product with itself is idempotent. (Contributed by Richard Penner, 24-Dec-2019.)
 |-  ( ph  ->  ( A  i^i  B )  =/=  (/) )   =>    |-  ( ph  ->  ( ( A  X.  B )  o.  ( A  X.  B ) )  =  ( A  X.  B ) )
 
Theoremxptrrel 37876 The cross product is always a transitive relation. (Contributed by Richard Penner, 24-Dec-2019.)
 |-  (
 ( A  X.  B )  o.  ( A  X.  B ) )  C_  ( A  X.  B )
 
Theorem0trrel 37877 The empty class is a transitive relation. (Contributed by Richard Penner, 24-Dec-2019.)
 |-  ( (/) 
 o.  (/) )  C_  (/)
 
Theoremtrrelssd 37878 The composition of subclasses of a transitive relation is a subclass of that relation. (Contributed by Richard Penner, 24-Dec-2019.)
 |-  ( ph  ->  ( R  o.  R )  C_  R )   &    |-  ( ph  ->  S  C_  R )   &    |-  ( ph  ->  T  C_  R )   =>    |-  ( ph  ->  ( S  o.  T )  C_  R )
 
Theoremtrrelind 37879 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 )
 
Theoremtrficl 37880* 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
 
Theoremxpintrreld 37881 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 37882 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 )
 
Theoremcnvtrrel 37883 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 )
 
Theoremtrrelsuperreldg 37884 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 ) )
 
21.32.2.2  Transitive closure of a relation
 
Theoremtrclub 37885* The Cartesian product of the domain and range of a relation is an upper bound for its transitive closure. (Contributed by Richard Penner, 2-Jan-2020.)
 |-  Rel  R   &    |-  R  e.  V   =>    |- 
 |^| { s  |  ( R  C_  s  /\  ( s  o.  s
 )  C_  s ) }  C_  ( dom  R  X.  ran  R )
 
Theoremtrclubg 37886* The union with the Cartesian product of its domain and range is an upper bound for a set's transitive closure. (Contributed by Richard Penner, 3-Jan-2020.)
 |-  R  e.  V   =>    |- 
 |^| { s  |  ( R  C_  s  /\  ( s  o.  s
 )  C_  s ) }  C_  ( R  u.  ( dom  R  X.  ran  R ) )
 
Theoremcotr2g 37887* Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr2 37888 for the main application. (Contributed by RP, 22-Mar-2020.)
 |-  dom  B 
 C_  D   &    |-  ( ran  B  i^i  dom  A )  C_  E   &    |- 
 ran  A  C_  F   =>    |-  ( ( A  o.  B )  C_  C 
 <-> 
 A. x  e.  D  A. y  e.  E  A. z  e.  F  (
 ( x B y 
 /\  y A z )  ->  x C z ) )
 
Theoremcotr2 37888* Two ways of saying a relation is transitive. Special instance of cotr2g 37887. (Contributed by RP, 22-Mar-2020.)
 |-  dom  R 
 C_  A   &    |-  ( dom  R  i^i  ran  R )  C_  B   &    |- 
 ran  R  C_  C   =>    |-  ( ( R  o.  R )  C_  R 
 <-> 
 A. x  e.  A  A. y  e.  B  A. z  e.  C  (
 ( x R y 
 /\  y R z )  ->  x R z ) )
 
Theoremcotr3 37889* Two ways of saying a relation is transitive. (Contributed by RP, 22-Mar-2020.)
 |-  A  =  dom  R   &    |-  B  =  ( A  i^i  C )   &    |-  C  =  ran  R   =>    |-  ( ( R  o.  R )  C_  R 
 <-> 
 A. x  e.  A  A. y  e.  B  A. z  e.  C  (
 ( x R y 
 /\  y R z )  ->  x R z ) )
 
21.32.3  Propositions from _Begriffsschrift_

Numbered propositions from [Frege1879]. ax-frege1 37914, ax-frege2 37916, ax-frege8 37937, ax-frege28 37958, ax-frege31 37962, ax-frege41 37973, frege52 (see ax-frege52a 37985, frege52b 38017, and ax-frege52c 38016 for translations), frege54 (see ax-frege54a 37990, frege54b 38021 and ax-frege54c 38020 for translations) and frege58 (see ax-frege58a 38003, ax-frege58b 38029 and frege58c 38049 for translations) are considered "core" or axioms. However, at least ax-frege8 37937 can be derived from ax-frege1 37914 and ax-frege2 37916, see axfrege8 37935.

English translations for specific propositions lifted in part from a translation by Stefan Bauer-Mengelberg as reprinted in From Frege to Goedel: A Source Book in Mathematical Logic, 1879-1931. An attempt to align these propositions in the larger Metamath dictionary has also been made.

 
21.32.3.1  _Begriffsschrift_ Chapter I

Section 2 introduces the turnstile  |- which turns an idea which may be true  ph into an assertion that it does hold true  |- 
ph. Section 5 introduces implication, 
( ph  ->  ps ). Section 6 introduces the single rule of interference relied upon, modus ponens ax-mp 5. Section 7 introduces negation and with in synonyms for or  ( -.  ph  ->  ps ), and  -.  ( ph  ->  -.  ps ), and two for exclusive-or corresponding to df-or 370, df-an 371, dfxor4 37890, dfxor5 37891.

Section 8 introduces the problematic notation for identity of conceptual content which must be separated into cases for biimplication  ( ph  <->  ps ) or class equality  A  =  B in this adaptation. Section 10 introduces "truth functions" for one or two variables in equally troubling notation, as the arguments may be understood to be logical predicates or collections. Here f( ph) is interpreted to mean if- ( ph ,  ps ,  ch ) where the content of the "function" is specified by the latter two argments or logical equivalent, while g( A) is read as  A  e.  G and h( A ,  B) as  A H B. This necessarily introduces a need for set theory as both  A  e.  G and  A H B cannot hold unless  A is a set. (Also  B.)

Section 11 introduces notation for generality, but there is no standard notation for generality when the variable is a proposition because it was realized after Frege that the universe of all possible propositions includes paradoxical constructions leading to the failure of naive set theory. So adopting f( ph) as if- ( ph ,  ps ,  ch ) would result in the translation of  A. ph f ( ph) as  ( ps 
/\  ch ). For collections, we must generalize over set variables or run into the same problems; this leads to  A. A g( A) being translated as  A. a a  e.  G and so forth.

Under this interpreation the text of section 11 gives us sp 1860 (or simpl 457 and simpr 461 and anifp 34259 in the propositional case) and statments similar to cbvalivw 1790, ax-gen 1619, alrimiv 1720, and alrimdv 1722. These last four introduce a generality and have no useful definition in terms of propositional variables.

Section 12 introduces some combinations of primitive symbols and their human language counterparts. Using class notation, these can also be expressed without dummy variables. All are A,  A. x x  e.  A,  -.  E. x -.  x  e.  A alex 1648, 
A  =  _V eqv 3810; Some are not B,  -.  A. x x  e.  B,  E. x -.  x  e.  B exnal 1649, 
B  C.  _V pssv 3869,  B  =/=  _V nev 37892; There are no C,  A. x -.  x  e.  C,  -.  E. x x  e.  C alnex 1615, 
C  =  (/) eq0 3809; There exist D,  -. 
A. x -.  x  e.  D,  E. x x  e.  D df-ex 1614,  (/)  C.  D 0pss 3867,  D  =/=  (/) n0 3803.

Notation for relations between expressions also can be written in various ways. All E are P,  A. x ( x  e.  E  ->  x  e.  P ),  -.  E. x
( x  e.  E  /\  -.  x  e.  P
) dfss6 37893, 
E  =  ( E  i^i  P ) df-ss 3485,  E  C_  P dfss2 3488; No F are P,  A. x ( x  e.  F  ->  -.  x  e.  P ),  -.  E. x
( x  e.  F  /\  x  e.  P
) alinexa 1664,  ( F  i^i  P
)  =  (/) disj1 3872; Some G are not P,  -.  A. x ( x  e.  G  ->  x  e.  P ),  E. x ( x  e.  G  /\  -.  x  e.  P
) exanali 1671,  ( G  i^i  P
)  C.  G nssinpss 3737,  -.  G  C_  P nss 3557; Some H are P,  -.  A. x
( x  e.  H  ->  -.  x  e.  P
),  E. x ( x  e.  H  /\  x  e.  P ) bj-exnalimn 34322,  (/)  C.  ( H  i^i  P
) 0pssin 37895, 
( H  i^i  P
)  =/=  (/) ndisj 37894.

 
Theoremdfxor4 37890 Express exclusive-or in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 14-Apr-2020.)
 |-  (
 ( ph  \/_  ps )  <->  -.  ( ( -.  ph  ->  ps )  ->  -.  ( ph  ->  -.  ps )
 ) )
 
Theoremdfxor5 37891 Express exclusive-or in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 14-Apr-2020.)
 |-  (
 ( ph  \/_  ps )  <->  -.  ( ( ph  ->  -. 
 ps )  ->  -.  ( -.  ph  ->  ps )
 ) )
 
Theoremnev 37892* Express that not every set is in a class. (Contributed by RP, 16-Apr-2020.)
 |-  ( A  =/=  _V  <->  -.  A. x  x  e.  A )
 
Theoremdfss6 37893* Another definition of subclasshood. (Contributed by RP, 16-Apr-2020.)
 |-  ( A  C_  B  <->  -.  E. x ( x  e.  A  /\  -.  x  e.  B ) )
 
Theoremndisj 37894* Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.)
 |-  (
 ( A  i^i  B )  =/=  (/)  <->  E. x ( x  e.  A  /\  x  e.  B ) )
 
Theorem0pssin 37895* Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.)
 |-  ( (/)  C.  ( A  i^i  B ) 
 <-> 
 E. x ( x  e.  A  /\  x  e.  B ) )
 
21.32.3.2  _Begriffsschrift_ Notation hints

The statement  R hereditary  A means relation  R is hereditary (in the sense of Frege) in the class  A or  ( R " A
)  C_  A. The former is only a slight reduction in the number of symbols, but this reduces the number of floating hypotheses needed to be checked.

As Frege wasn't using the language of classes or sets, this naturally differs from the set-theoretic notion that a set is hereditary in a property: that all of its elements have a property and all of their elements have the property and so-on.

 
Theoremrp-imass 37896 If the  R-image of a class  A is a subclass of  B, then the restriction of  R to  A is a subset of the Cartesian product of  A and  B. (Contributed by Richard Penner, 24-Dec-2019.)
 |-  (
 ( R " A )  C_  B  <->  ( R  |`  A ) 
 C_  ( A  X.  B ) )
 
Syntaxwhe 37897 The property of relation  R being hereditary in class  A.
 wff  R hereditary  A
 
Definitiondf-he 37898 The property of relation  R being hereditary in class  A. (Contributed by RP, 27-Mar-2020.)
 |-  ( R hereditary  A  <->  ( R " A )  C_  A )
 
Theoremdfhe2 37899 The property of relation  R being hereditary in class  A. (Contributed by RP, 27-Mar-2020.)
 |-  ( R hereditary  A  <->  ( R  |`  A ) 
 C_  ( A  X.  A ) )
 
Theoremdfhe3 37900* The property of relation  R being hereditary in class  A. (Contributed by RP, 27-Mar-2020.)
 |-  ( R hereditary  A  <->  A. x ( x  e.  A  ->  A. y
 ( x R y 
 ->  y  e.  A ) ) )
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