HomeHome Metamath Proof Explorer
Theorem List (p. 383 of 386)
< 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
(1-26036)
  Hilbert Space Explorer  Hilbert Space Explorer
(26037-27561)
  Users' Mathboxes  Users' Mathboxes
(27562-38552)
 

Theorem List for Metamath Proof Explorer - 38201-38300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremintimass 38201* 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 38202* 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 38203* 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 38204* 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 38205* 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 }
 ) )
 
21.31.2.1  Transitive relations (not to be confused with transitive classes).
 
Theoremtrrelind 38206 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 38207* 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 38208 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 38209 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 38210 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 38211 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 ) )
 
21.31.2.2  Reflexive closures
 
Syntaxcrcl 38212 Extend class notation with reflexive closure.
 class  r*
 
Definitiondf-rcl 38213* Reflexive closure of a relation. This is the smallest superset which has the reflexive property. (Contributed by RP, 5-Jun-2020.)
 |-  r*  =  ( x  e.  _V  |->  |^| { z  |  ( x  C_  z  /\  (  _I  |`  ( dom  z  u.  ran  z
 ) )  C_  z
 ) } )
 
Theoremdfrcl2 38214 Reflexive closure of a relation as union with restricted identity relation. (Contributed by RP, 6-Jun-2020.)
 |-  r*  =  ( x  e.  _V  |->  ( (  _I  |`  ( dom  x  u.  ran 
 x ) )  u.  x ) )
 
Theoremdfrcl3 38215 Reflexive closure of a relation as union of powers of the relation. (Contributed by RP, 6-Jun-2020.)
 |-  r*  =  ( x  e.  _V  |->  ( ( x ^r  0 )  u.  ( x ^r  1 ) ) )
 
Theoremdfid5 38216 Identity relation is equal to relational exponentiation to the first power. (Contributed by RP, 9-Jun-2020.)
 |-  _I  =  ( x  e.  _V  |->  ( x ^r 
 1 ) )
 
Theoremdfid6 38217* Identity relation expressed as indexed union of relational powers. (Contributed by RP, 9-Jun-2020.)
 |-  _I  =  ( x  e.  _V  |->  U_ n  e.  { 1 }  ( x ^r  n ) )
 
Theoremdfrcl4 38218* Reflexive closure of a relation as indexed union of powers of the relation. (Contributed by RP, 8-Jun-2020.)
 |-  r*  =  ( r  e.  _V  |->  U_ n  e.  {
 0 ,  1 }  ( r ^r  n ) )
 
21.31.2.3  Finite relationship composition.

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

 
Theoremrelexp2 38219 A set operated on by the relation exponent to the second power is equal to the composition of the set with itself. (Contributed by RP, 1-Jun-2020.)
 |-  ( R  e.  V  ->  ( R ^r  2 )  =  ( R  o.  R ) )
 
Theoremrelexpnul 38220 If the domain and range of powers of a relation are disjoint then the relation raised to the sum of those exponents is empty. (Contributed by RP, 1-Jun-2020.)
 |-  (
 ( ( R  e.  V  /\  Rel  R )  /\  ( N  e.  NN0  /\  M  e.  NN0 )
 )  ->  ( ( dom  ( R ^r  N )  i^i  ran  ( R ^r  M ) )  =  (/)  <->  ( R ^r  ( N  +  M ) )  =  (/) ) )
 
Theoremeliunov2 38221* Membership in the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the the element is a member of that operator value. Generalized from dfrtrclrec2 12975. (Contributed by RP, 1-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  N  ( r  .^  n )
 )   =>    |-  ( ( R  e.  U  /\  N  e.  V )  ->  ( X  e.  ( C `  R )  <->  E. n  e.  N  X  e.  ( R  .^  n ) ) )
 
Theoremov2ssiunov2 38222* Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 12976 and rtrclreclem2 . (Contributed by RP, 4-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  N  ( r  .^  n )
 )   =>    |-  ( ( R  e.  U  /\  N  e.  V  /\  M  e.  N ) 
 ->  ( R  .^  M )  C_  ( C `  R ) )
 
Theorembriunov2 38223* Two classes related by the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the the two classes are related by that operator value. (Contributed by RP, 1-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  N  ( r  .^  n )
 )   =>    |-  ( ( R  e.  U  /\  N  e.  V )  ->  ( X ( C `  R ) Y  <->  E. n  e.  N  X ( R  .^  n ) Y ) )
 
Theoremeliunov2uz 38224* Membership in the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the the element is a member of that operator value. The index set  N is restricted to an upper range of integers. (Contributed by RP, 2-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  N  ( r  .^  n )
 )   =>    |-  ( ( R  e.  U  /\  N  =  (
 ZZ>= `  M ) ) 
 ->  ( X  e.  ( C `  R )  <->  E. n  e.  N  X  e.  ( R  .^  n ) ) )
 
Theorembriunov2uz 38225* Two classes related by the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the the two classes are related by that operator value. The index set  N is restricted to an upper range of integers. (Contributed by RP, 2-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  N  ( r  .^  n )
 )   =>    |-  ( ( R  e.  U  /\  N  =  (
 ZZ>= `  M ) ) 
 ->  ( X ( C `
  R ) Y  <->  E. n  e.  N  X ( R  .^  n ) Y ) )
 
Theoremrelexpaddss 38226 The composition of two powers of a relation is a subset of the relation raised to the sum of those exponents. This is equality where  R is a relation as shown by relexpaddd 12972 or when the sum of the powers isn't 1 as shown by relexpaddg 12971. (Contributed by RP, 3-Jun-2020.)
 |-  (
 ( N  e.  NN0  /\  M  e.  NN0  /\  R  e.  V )  ->  (
 ( R ^r  N )  o.  ( R ^r  M ) )  C_  ( R ^r  ( N  +  M ) ) )
 
Theoremiunrelexpuztr 38227* The indexed union of relation exponentiation over upper integers is a transive relation. Generalized from rtrclreclem3 12978. (Contributed by RP, 4-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  N  ( r ^r  n ) )   =>    |-  ( ( R  e.  V  /\  N  =  (
 ZZ>= `  M )  /\  M  e.  NN0 )  ->  ( ( C `  R )  o.  ( C `  R ) ) 
 C_  ( C `  R ) )
 
Theoremiunrelexpmin1 38228* The indexed union of relation exponentiation over the natural numbers is the minimum transitive relation that includes the relation. (Contributed by RP, 4-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  N  ( r ^r  n ) )   =>    |-  ( ( R  e.  V  /\  N  =  NN )  ->  A. s ( ( R  C_  s  /\  ( s  o.  s
 )  C_  s )  ->  ( C `  R )  C_  s ) )
 
Theoremiunrelexpmin2 38229* The indexed union of relation exponentiation over the natural numbers (including zero) is the minimum reflexive-transitive relation that includes the relation. (Contributed by RP, 4-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  N  ( r ^r  n ) )   =>    |-  ( ( R  e.  V  /\  N  =  NN0 )  ->  A. s ( ( (  _I  |`  ( dom 
 R  u.  ran  R ) )  C_  s  /\  R  C_  s  /\  (
 s  o.  s ) 
 C_  s )  ->  ( C `  R ) 
 C_  s ) )
 
Theoremeltrclrec 38230* Membership in the indexed union of relation exponentiation over the natural numbers is equivalent to the existence of at least one number such that the element is a member of that relationship power. (Contributed by RP, 2-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  NN  (
 r ^r  n ) )   =>    |-  ( R  e.  V  ->  ( X  e.  ( C `  R )  <->  E. n  e.  NN  X  e.  ( R ^r  n ) ) )
 
Theorembrtrclrec 38231* Two classes related by the indexed union of relation exponentiation over the natural numbers is equivalent to the existence of at least one number such that the two classes are related by that relationship power. (Contributed by RP, 2-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  NN  (
 r ^r  n ) )   =>    |-  ( R  e.  V  ->  ( X ( C `
  R ) Y  <->  E. n  e.  NN  X ( R ^r  n ) Y ) )
 
Theoremelrtrclrec 38232* Membership in the indexed union of relation exponentiation over the natural numbers (including zero) is equivalent to the existence of at least one number such that the element is a member of that relationship power. (Contributed by RP, 2-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  NN0  (
 r ^r  n ) )   =>    |-  ( R  e.  V  ->  ( X  e.  ( C `  R )  <->  E. n  e.  NN0  X  e.  ( R ^r  n ) ) )
 
Theorembrrtrclrec 38233* Two classes related by the indexed union of relation exponentiation over the natural numbers (including zero) is equivalent to the existence of at least one number such that the two classes are related by that relationship power. (Contributed by RP, 2-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  NN0  (
 r ^r  n ) )   =>    |-  ( R  e.  V  ->  ( X ( C `
  R ) Y  <->  E. n  e.  NN0  X ( R ^r  n ) Y ) )
 
Theoremdftrcl3 38234* Transitive closure of a relation, expressed as indexed union of powers of relations. (Contributed by RP, 5-Jun-2020.)
 |-  t+  =  ( r  e.  _V  |->  U_ n  e.  NN  ( r ^r  n ) )
 
Theoremdfrtrcl3 38235* Reflexive-transitive closure of a relation, expressed as indexed union of powers of relations. Generalized from dfrtrcl2 12980. (Contributed by RP, 5-Jun-2020.)
 |-  t*  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^r  n ) )
 
Theoremdfrtrcl4 38236 Reflexive-transitive closure of a relation, expressed as the union of the zeroth power and the transitive closure. (Contributed by RP, 5-Jun-2020.)
 |-  t*  =  ( r  e.  _V  |->  ( ( r ^r  0 )  u.  ( t+ `
  r ) ) )
 
Theoremcomptiunov2i 38237* The composition two indexed unions is sometimes a similar indexed union. (Contributed by RP, 10-Jun-2020.)
 |-  X  =  ( a  e.  _V  |->  U_ i  e.  I  ( a  .^  i )
 )   &    |-  Y  =  ( b  e.  _V  |->  U_ j  e.  J  ( b  .^  j ) )   &    |-  Z  =  ( c  e.  _V  |->  U_ k  e.  K  ( c  .^  k )
 )   &    |-  I  e.  _V   &    |-  J  e.  _V   &    |-  K  =  ( I  u.  J )   &    |-  U_ k  e.  I  ( d  .^  k )  C_  U_ i  e.  I  ( U_ j  e.  J  ( d  .^  j ) 
 .^  i )   &    |-  U_ k  e.  J  ( d  .^  k )  C_  U_ i  e.  I  ( U_ j  e.  J  (
 d  .^  j )  .^  i )   &    |-  U_ i  e.  I  ( U_ j  e.  J  ( d  .^  j ) 
 .^  i )  C_  U_ k  e.  ( I  u.  J ) ( d  .^  k )   =>    |-  ( X  o.  Y )  =  Z
 
Theoremrelexpss1d 38238 The relational power of a subset is a subset. (Contributed by RP, 17-Jun-2020.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( A ^r  N ) 
 C_  ( B ^r  N ) )
 
Theoremrelexp0eq 38239 The zeroth power of relationships is the same if and only if the union of their domain and ranges is the same. (Contributed by RP, 11-Jun-2020.)
 |-  (
 ( A  e.  U  /\  B  e.  V ) 
 ->  ( ( dom  A  u.  ran  A )  =  ( dom  B  u.  ran 
 B )  <->  ( A ^r  0 )  =  ( B ^r 
 0 ) ) )
 
Theoremrelexp01min 38240 With exponents limited to 0 and 1, the composition of powers of a relation is the relation raised to the minimum of exponents. (Contributed by RP, 12-Jun-2020.)
 |-  (
 ( ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
 )  /\  ( J  e.  { 0 ,  1 }  /\  K  e.  { 0 ,  1 } ) )  ->  (
 ( R ^r  J ) ^r  K )  =  ( R ^r  I ) )
 
Theoremrelexp0idm 38241 Repeated raising a relation to the zeroth power is idempotent. (Contributed by RP, 12-Jun-2020.)
 |-  ( R  e.  V  ->  ( ( R ^r 
 0 ) ^r 
 0 )  =  ( R ^r  0 ) )
 
Theoremrelexp1idm 38242 Repeated raising a relation to the first power is idempotent. (Contributed by RP, 12-Jun-2020.)
 |-  ( R  e.  V  ->  ( ( R ^r 
 1 ) ^r 
 1 )  =  ( R ^r  1 ) )
 
Theoremrelexpiidm 38243 Any power of any restriction of the identity relation is itself. (Contributed by RP, 12-Jun-2020.)
 |-  (
 ( A  e.  V  /\  N  e.  NN0 )  ->  ( (  _I  |`  A ) ^r  N )  =  (  _I  |`  A ) )
 
Theoremrelexp0a 38244 Absorbtion law for zeroth power of a relation. (Contributed by RP, 17-Jun-2020.)
 |-  (
 ( A  e.  V  /\  N  e.  NN0 )  ->  ( ( A ^r  N ) ^r 
 0 )  C_  ( A ^r  0 ) )
 
Theoremiunrelexp0 38245* Simplification of zeroth power of indexed union of powers of relations. (Contributed by RP, 19-Jun-2020.)
 |-  (
 ( R  e.  V  /\  Z  C_  NN0  /\  ( { 0 ,  1 }  i^i  Z )  =/=  (/) )  ->  ( U_ x  e.  Z  ( R ^r  x ) ^r 
 0 )  =  ( R ^r  0 ) )
 
Theoremrelexpxpnnidm 38246 Any positive power of a cross product of non-disjoint sets is itself. (Contributed by RP, 13-Jun-2020.)
 |-  ( N  e.  NN  ->  ( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) )  ->  (
 ( A  X.  B ) ^r  N )  =  ( A  X.  B ) ) )
 
Theoremrelexpxpmin 38247 The composition of powers of a cross-product of non-disjoint sets is the cross product raised to the minimum exponent. (Contributed by RP, 13-Jun-2020.)
 |-  (
 ( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) )  /\  ( I  =  if ( J  <  K ,  J ,  K )  /\  J  e.  NN0  /\  K  e.  NN0 ) )  ->  (
 ( ( A  X.  B ) ^r  J ) ^r  K )  =  (
 ( A  X.  B ) ^r  I ) )
 
Theoremrelexpmulnn 38248 With exponents limited to the counting numbers, the composition of powers of a relation is the relation raised to the product of exponents. (Contributed by RP, 13-Jun-2020.)
 |-  (
 ( ( R  e.  V  /\  I  =  ( J  x.  K ) )  /\  ( J  e.  NN  /\  K  e.  NN ) )  ->  ( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) )
 
Theoremrelexpmulg 38249 With ordered exponents, the composition of powers of a relation is the relation raised to the product of exponents. (Contributed by RP, 13-Jun-2020.)
 |-  (
 ( ( R  e.  V  /\  I  =  ( J  x.  K ) 
 /\  ( I  =  0  ->  J  <_  K ) )  /\  ( J  e.  NN0  /\  K  e.  NN0 ) )  ->  ( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) )
 
Theoremtrclrelexplem 38250* The union of relational powers to positive multiples of  N is a subset to the transitive closure raised to the power of  N. (Contributed by RP, 15-Jun-2020.)
 |-  ( N  e.  NN  ->  U_ k  e.  NN  (
 ( D ^r 
 k ) ^r  N )  C_  ( U_ j  e.  NN  ( D ^r  j ) ^r  N ) )
 
Theoremcorclrcl 38251 The reflexive closure is idempotent. (Contributed by RP, 13-Jun-2020.)
 |-  (
 r*  o.  r* )  =  r*
 
Theoremcorcltrcl 38252 The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 17-Jun-2020.)
 |-  (
 r*  o.  t+ )  =  t*
 
Theoremcotrclrcl 38253 The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 21-Jun-2020.)
 |-  (
 t+  o.  r* )  =  t*
 
Theoremcotrcltrcl 38254 The transitive closure is idempotent. (Contributed by RP, 16-Jun-2020.)
 |-  (
 t+  o.  t+ )  =  t+
 
Theoremcorclrtrcl 38255 Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.)
 |-  (
 r*  o.  t* )  =  t*
 
Theoremcotrclrtrcl 38256 Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.)
 |-  (
 t+  o.  t* )  =  t*
 
Theoremcortrclrcl 38257 Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.)
 |-  (
 t*  o.  r* )  =  t*
 
Theoremcortrcltrcl 38258 Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.)
 |-  (
 t*  o.  t+ )  =  t*
 
Theoremcortrclrtrcl 38259 The reflexive-transitive closure is idempotent. (Contributed by RP, 13-Jun-2020.)
 |-  (
 t*  o.  t* )  =  t*
 
Theoremtrclimalb2 38260 Lower bound for image under a transitive closure. (Contributed by RP, 1-Jul-2020.)
 |-  (
 ( R  e.  V  /\  ( R " ( A  u.  B ) ) 
 C_  B )  ->  ( ( t+ `
  R ) " A )  C_  B )
 
21.31.2.4  Transitive closure of a relation
 
Theorembrtrclfv2 38261* Two ways to indicate two elements are related by the transitive closure of a relation. (Contributed by RP, 1-Jul-2020.)
 |-  (
 ( X  e.  U  /\  Y  e.  V  /\  R  e.  W )  ->  ( X ( t+ `  R ) Y  <->  Y  e.  |^| { f  |  ( R " ( { X }  u.  f
 ) )  C_  f } ) )
 
21.31.3  Propositions from _Begriffsschrift_

Numbered propositions from [Frege1879]. ax-frege1 38287, ax-frege2 38289, ax-frege8 38310, ax-frege28 38331, ax-frege31 38335, ax-frege41 38346, frege52 (see ax-frege52a 38358, frege52b 38390, and ax-frege52c 38389 for translations), frege54 (see ax-frege54a 38363, frege54b 38394 and ax-frege54c 38393 for translations) and frege58 (see ax-frege58a 38376, ax-frege58b 38402 and frege58c 38422 for translations) are considered "core" or axioms. However, at least ax-frege8 38310 can be derived from ax-frege1 38287 and ax-frege2 38289, see axfrege8 38308.

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.

 
21.31.3.1  _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 368, df-an 369, dfxor4 38262, dfxor5 38263.

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 1864 (or simpl 455 and simpr 459 and anifp 34578 in the propositional case) and statments similar to cbvalivw 1794, ax-gen 1623, alrimiv 1724, and alrimdv 1726. 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 1652, 
A  =  _V eqv 3800; Some are not B,  -.  A. x x  e.  B,  E. x -.  x  e.  B exnal 1653, 
B  C.  _V pssv 3854,  B  =/=  _V nev 38264; There are no C,  A. x -.  x  e.  C,  -.  E. x x  e.  C alnex 1619, 
C  =  (/) eq0 3799; There exist D,  -. 
A. x -.  x  e.  D,  E. x x  e.  D df-ex 1618,  (/)  C.  D 0pss 3852,  D  =/=  (/) n0 3793.

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 38265, 
E  =  ( E  i^i  P ) df-ss 3475,  E  C_  P dfss2 3478; No F are P,  A. x ( x  e.  F  ->  -.  x  e.  P ),  -.  E. x
( x  e.  F  /\  x  e.  P
) alinexa 1668,  ( F  i^i  P
)  =  (/) disj1 3857; Some G are not P,  -.  A. x ( x  e.  G  ->  x  e.  P ),  E. x ( x  e.  G  /\  -.  x  e.  P
) exanali 1675,  ( G  i^i  P
)  C.  G nssinpss 3727,  -.  G  C_  P nss 3547; Some H are P,  -.  A. x
( x  e.  H  ->  -.  x  e.  P
),  E. x ( x  e.  H  /\  x  e.  P ) bj-exnalimn 34641,  (/)  C.  ( H  i^i  P
) 0pssin 38267, 
( H  i^i  P
)  =/=  (/) ndisj 38266.

 
Theoremdfxor4 38262 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 38263 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 38264* Express that not every set is in a class. (Contributed by RP, 16-Apr-2020.)
 |-  ( A  =/=  _V  <->  -.  A. x  x  e.  A )
 
Theoremdfss6 38265* Another definition of subclasshood. (Contributed by RP, 16-Apr-2020.)
 |-  ( A  C_  B  <->  -.  E. x ( x  e.  A  /\  -.  x  e.  B ) )
 
Theoremndisj 38266* 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 38267* 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 ) )
 
21.31.3.2  _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 38268 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 38269 The property of relation  R being hereditary in class  A.
 wff  R hereditary  A
 
Definitiondf-he 38270 The property of relation  R being hereditary in class  A. (Contributed by RP, 27-Mar-2020.)
 |-  ( R hereditary  A  <->  ( R " A )  C_  A )
 
Theoremdfhe2 38271 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 38272* 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 ) ) )
 
Theoremheeq12 38273 Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
 |-  (
 ( R  =  S  /\  A  =  B ) 
 ->  ( R hereditary  A  <->  S hereditary  B ) )
 
Theoremheeq1 38274 Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
 |-  ( R  =  S  ->  ( R hereditary  A  <->  S hereditary  A ) )
 
Theoremheeq2 38275 Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
 |-  ( A  =  B  ->  ( R hereditary  A  <->  R hereditary  B ) )
 
Theoremsbcheg 38276 Distribute proper substitution through herditary relation. (Contributed by RP, 29-Jun-2020.)
 |-  ( A  e.  V  ->  (
 [. A  /  x ]. B hereditary  C  <->  [_ A  /  x ]_ B hereditary  [_ A  /  x ]_ C ) )
 
Theoremhess 38277 Subclass law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
 |-  ( S  C_  R  ->  ( R hereditary  A  ->  S hereditary  A ) )
 
Theoremxphe 38278 Any Cartesian product is hereditary in its second class. (Contributed by RP, 27-Mar-2020.)
 |-  ( A  X.  B ) hereditary  B
 
Theorem0he 38279 The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.)
 |-  (/) hereditary  A
 
Theorem0heALT 38280 The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  (/) hereditary  A
 
Theoremhe0 38281 Any relation is hereditary in the empty set. (Contributed by RP, 27-Mar-2020.)
 |-  A hereditary  (/)
 
Theoremunhe1 38282 The union of two relations hereditary in a class is also hereditary in a class. (Contributed by RP, 28-Mar-2020.)
 |-  (
 ( R hereditary  A  /\  S hereditary  A )  ->  ( R  u.  S ) hereditary  A )
 
Theoremsnhesn 38283 Any singleton is hereditary in any singleton. (Contributed by RP, 28-Mar-2020.)
 |-  { <. A ,  A >. } hereditary  { B }
 
Theoremidhe 38284 The identity relation is hereditary in any class. (Contributed by RP, 28-Mar-2020.)
 |-  _I hereditary  A
 
Theorempsshepw 38285 The relation between sets and their proper subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.)
 |-  `' [ C.] hereditary  ~P A
 
Theoremsshepw 38286 The relation between sets and their subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.)
 |-  ( `' [ C.]  u.  _I  ) hereditary  ~P A
 
21.31.3.3  _Begriffsschrift_ Chapter II Implication
 
Axiomax-frege1 38287 The case in which  ph is denied,  ps is affirmed, and 
ph is affirmed is excluded. This is evident since  ph cannot at the same time be denied and affirmed. Axiom 1 of [Frege1879] p. 26. Identical to ax-1 6. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ph ) )
 
Theoremrp-a1i 38288 Identical to a1i 11. (Contributed by RP, 24-Dec-2019.)
 |-  ps   =>    |-  ( ph  ->  ps )
 
Axiomax-frege2 38289 If a proposition  ch is a necessary consequence of two propositions  ps and  ph and one of those,  ps, is in turn a necessary consequence of the other, 
ph, then the proposition  ch is a necessary consequence of the latter one,  ph, alone. Axiom 2 of [Frege1879] p. 26. Identical to ax-2 7. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ( ps 
 ->  ch ) )  ->  ( ( ph  ->  ps )  ->  ( ph  ->  ch ) ) )
 
Theoremrp-frege2i 38290 Identical to a2i 13. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ( ph  ->  ps )  ->  ( ph  ->  ch ) )
 
Theoremrp-frege2ii 38291 Identical to mpd 15. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  ( ph  ->  ch )   &    |-  ( ph  ->  ( ch  ->  ps )
 )   =>    |-  ( ph  ->  ps )
 
Theoremrp-frege2iii 38292 Implication derived from ax-frege2 38289. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  ps   &    |-  ( ps  ->  ch )   &    |-  ( ps  ->  ( ch  ->  ph ) )   =>    |-  ph
 
Theoremrp-simp2-frege 38293 Simplification of triple conjunction. Compare with simp2 995. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ps )
 ) )
 
Theoremrp-simp2 38294 Simplification of triple conjunction. Identical to simp2 995. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  ps )
 
Theoremrp-frege3g 38295 Add antecedent to ax-frege2 38289. More general statement than frege3 38296. Like ax-frege2 38289, it is essentially a closed form of mpd 15, however it has an extra antecedent.

It would be more natural to prove from a1i 11 and ax-frege2 38289 in Metamath. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

 |-  ( ph  ->  ( ( ps 
 ->  ( ch  ->  th )
 )  ->  ( ( ps  ->  ch )  ->  ( ps  ->  th ) ) ) )
 
Theoremfrege3 38296 Add antecedent to ax-frege2 38289. Special case of rp-frege3g 38295. Proposition 3 of [Frege1879] p. 29. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ph  ->  ps )  ->  ( ( ch  ->  (
 ph  ->  ps ) )  ->  ( ( ch  ->  ph )  ->  ( ch  ->  ps ) ) ) )
 
Theoremrp-misc1-frege 38297 Double-use of ax-frege2 38289. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ( ph  ->  ( ps  ->  ch )
 )  ->  ( ph  ->  ps ) )  ->  ( ( ph  ->  ( ps  ->  ch )
 )  ->  ( ph  ->  ch ) ) )
 
Theoremrp-frege24 38298 Introducing an embedded antecedent. Alternate proof for frege24 38316. Closed form for a1d 25. (Contributed by RP, 24-Dec-2019.)
 |-  (
 ( ph  ->  ps )  ->  ( ph  ->  ( ch  ->  ps ) ) )
 
Theoremrp-frege4g 38299 Deduction relatied to distribution. (Contributed by RP, 24-Dec-2019.)
 |-  (
 ( ph  ->  ( ps 
 ->  ( ch  ->  th )
 ) )  ->  ( ph  ->  ( ( ps 
 ->  ch )  ->  ( ps  ->  th ) ) ) )
 
Theoremfrege4 38300 Special case of closed form of a2d 26. Special case of rp-frege4g 38299. Proposition 4 of [Frege1879] p. 31. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ( ph  ->  ps )  ->  ( ch  ->  ( ph  ->  ps )
 ) )  ->  (
 ( ph  ->  ps )  ->  ( ( ch  ->  ph )  ->  ( ch  ->  ps ) ) ) )
    < 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 38501-38552
  Copyright terms: Public domain < Previous  Next >