P. 383 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 38201-38300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	frege62b 38201	A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2563 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( [ x / y ] ph -> ( A. y ( ph -> ps ) -> [ x / y ] ps ) )
Theorem	frege63b 38202	Lemma for frege91 38248. Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( [ x / y ] ph -> ( ps -> ( A. y ( ph -> ch ) -> [ x / y ] ch ) ) )
Theorem	frege64b 38203	Lemma for frege65b 38204. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( [ x / y ] ph -> [ z / y ] ps ) -> ( A. y ( ps -> ch ) -> ( [ x / y ] ph -> [ z / y ] ch ) ) )
Theorem	frege65b 38204	A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2563 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53. In Frege care is taken to point out that the variables in the first clauses are independent of each other and of the final term so another valid translation could be : |- ( A. x ( [ x / a ] ph -> [ x / b ] ps ) -> ( A. y ( [ y / b ] ps -> [ y / c ] ch ) -> ( [ z / a ] ph -> [ z / c ] ch ) ) ). But that is perhaps too pedantic a translation for this exploration. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( A. x ( ph -> ps ) -> ( A. x ( ps -> ch ) -> ( [ y / x ] ph -> [ y / x ] ch ) ) )
Theorem	frege66b 38205	Swap antecedents of frege65b 38204. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( A. x ( ph -> ps ) -> ( A. x ( ch -> ph ) -> ( [ y / x ] ch -> [ y / x ] ps ) ) )
Theorem	frege67b 38206	Lemma for frege68b 38207. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ( A. x ph <-> ps ) -> ( ps -> A. x ph ) ) -> ( ( A. x ph <-> ps ) -> ( ps -> [ y / x ] ph ) ) )
Theorem	frege68b 38207	Combination of applying a definition and applying it to a specific instance. Proposition 68 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( A. x ph <-> ps ) -> ( ps -> [ y / x ] ph ) )
20.27.3.7  _Begriffsschrift_ Chapter II with equivalence of classes (where they are sets)
Theorem	frege53c 38208	Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( [. A / x ]. ph -> ( A = B -> [. B / x ]. ph ) )
Theorem	frege54cor1c 38209*	Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Revised by RP, 25-Apr-2020.)
|- A e. C   =>    |- [. A / x ]. x = A
Theorem	frege55lem1c 38210*	Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.)
|- ( ( ph -> [. A / x ]. x = B ) -> ( ph -> A = B ) )
Theorem	frege55lem2c 38211*	Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( x = A -> [. A / z ]. z = x )
Theorem	frege55c 38212	Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( x = A -> A = x )
Theorem	frege56c 38213*	Lemma for frege57c 38214. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- B e. C   =>    |- ( ( A = B -> ( [. A / x ]. ph -> [. B / x ]. ph ) ) -> ( B = A -> ( [. A / x ]. ph -> [. B / x ]. ph ) ) )
Theorem	frege57c 38214*	Swap order of implication in ax-frege52c 38182. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- A e. C   =>    |- ( A = B -> ( [. B / x ]. ph -> [. A / x ]. ph ) )
Theorem	frege58c 38215	Principle related to sp 2053. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- A e. B   =>    |- ( A. x ph -> [. A / x ]. ph )
Theorem	frege59c 38216	A kind of Aristotelian inference. Proposition 59 of [Frege1879] p. 51. Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the frege12 38107 incorrectly referenced where frege30 38126 is in the original. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- A e. B   =>    |- ( [. A / x ]. ph -> ( -. [. A / x ]. ps -> -. A. x ( ph -> ps ) ) )
Theorem	frege60c 38217	Swap antecedents of frege58c 38215. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- A e. B   =>    |- ( A. x ( ph -> ( ps -> ch ) ) -> ( [. A / x ]. ps -> ( [. A / x ]. ph -> [. A / x ]. ch ) ) )
Theorem	frege61c 38218	Lemma for frege65c 38222. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- A e. B   =>    |- ( ( [. A / x ]. ph -> ps ) -> ( A. x ph -> ps ) )
Theorem	frege62c 38219	A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2563 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- A e. B   =>    |- ( [. A / x ]. ph -> ( A. x ( ph -> ps ) -> [. A / x ]. ps ) )
Theorem	frege63c 38220	Analogue of frege63b 38202. Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- A e. B   =>    |- ( [. A / x ]. ph -> ( ps -> ( A. x ( ph -> ch ) -> [. A / x ]. ch ) ) )
Theorem	frege64c 38221	Lemma for frege65c 38222. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- A e. B   =>    |- ( ( [. C / x ]. ph -> [. A / x ]. ps ) -> ( A. x ( ps -> ch ) -> ( [. C / x ]. ph -> [. A / x ]. ch ) ) )
Theorem	frege65c 38222	A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2563 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- A e. B   =>    |- ( A. x ( ph -> ps ) -> ( A. x ( ps -> ch ) -> ( [. A / x ]. ph -> [. A / x ]. ch ) ) )
Theorem	frege66c 38223	Swap antecedents of frege65c 38222. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- A e. B   =>    |- ( A. x ( ph -> ps ) -> ( A. x ( ch -> ph ) -> ( [. A / x ]. ch -> [. A / x ]. ps ) ) )
Theorem	frege67c 38224	Lemma for frege68c 38225. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- A e. B   =>    |- ( ( ( A. x ph <-> ps ) -> ( ps -> A. x ph ) ) -> ( ( A. x ph <-> ps ) -> ( ps -> [. A / x ]. ph ) ) )
Theorem	frege68c 38225	Combination of applying a definition and applying it to a specific instance. Proposition 68 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- A e. B   =>    |- ( ( A. x ph <-> ps ) -> ( ps -> [. A / x ]. ph ) )
20.27.3.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence ( R " A ) C_ A means membership in A is hereditary in the sequence dictated by relation R. This 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. While the above notation is modern, it is cumbersome in the case when A is complex and to more closely follow Frege, we abbreviate it with new notation R hereditary A. This greatly shortens the statements for frege97 38254 and frege109 38266. dffrege69 38226 through frege75 38232 develop this, but translation to Metamath is pending some decisions. While Frege does not limit discussion to sets, we may have to depart from Frege by limiting R or A to sets when we quantify over all hereditary relations or all classes where membership is hereditary in a sequence dictated by R.
Theorem	dffrege69 38226*	If from the proposition that x has property A it can be inferred generally, whatever x may be, that every result of an application of the procedure R to x has property A, then we say " Property A is hereditary in the R-sequence. Definition 69 of [Frege1879] p. 55. (Contributed by RP, 28-Mar-2020.)
|- ( A. x ( x e. A -> A. y ( x R y -> y e. A ) ) <-> R hereditary A )
Theorem	frege70 38227*	Lemma for frege72 38229. Proposition 70 of [Frege1879] p. 58. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.)
|- X e. V   =>    |- ( R hereditary A -> ( X e. A -> A. y ( X R y -> y e. A ) ) )
Theorem	frege71 38228*	Lemma for frege72 38229. Proposition 71 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.)
|- X e. V   =>    |- ( ( A. z ( X R z -> z e. A ) -> ( X R Y -> Y e. A ) ) -> ( R hereditary A -> ( X e. A -> ( X R Y -> Y e. A ) ) ) )
Theorem	frege72 38229	If property A is hereditary in the R-sequence, if x has property A, and if y is a result of an application of the procedure R to x, then y has property A. Proposition 72 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   =>    |- ( R hereditary A -> ( X e. A -> ( X R Y -> Y e. A ) ) )
Theorem	frege73 38230	Lemma for frege87 38244. Proposition 73 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   =>    |- ( ( R hereditary A -> X e. A ) -> ( R hereditary A -> ( X R Y -> Y e. A ) ) )
Theorem	frege74 38231	If X has a property A that is hereditary in the R-sequence, then every result of a application of the procedure R to X has the property A. Proposition 74 of [Frege1879] p. 60. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   =>    |- ( X e. A -> ( R hereditary A -> ( X R Y -> Y e. A ) ) )
Theorem	frege75 38232*	If from the proposition that x has property A, whatever x may be, it can be inferred that every result of an application of the procedure R to x has property A, then property A is hereditary in the R-sequence. Proposition 75 of [Frege1879] p. 60. (Contributed by RP, 28-Mar-2020.) (Proof modification is discouraged.)
|- ( A. x ( x e. A -> A. y ( x R y -> y e. A ) ) -> R hereditary A )
20.27.3.9  _Begriffsschrift_ Chapter III Following in a sequence p ( t+ ` R ) c means c follows p in the R-sequence. dffrege76 38233 through frege98 38255 develop this. This will be shown to be the transitive closure of the relation R. But more work needs to be done on transitive closure of relations before this is ready for Metamath.
Theorem	dffrege76 38233*	If from the two propositions that every result of an application of the procedure R to B has property f and that property f is hereditary in the R-sequence, it can be inferred, whatever f may be, that E has property f, then we say E follows B in the R-sequence. Definition 76 of [Frege1879] p. 60. Each of B, E and R must be sets. (Contributed by RP, 2-Jul-2020.)
|- B e. U   &    |- E e. V   &    |- R e. W   =>    |- ( A. f ( R hereditary f -> ( A. a ( B R a -> a e. f ) -> E e. f ) ) <-> B ( t+ ` R ) E )
Theorem	frege77 38234*	If Y follows X in the R-sequence, if property A is hereditary in the R-sequence, and if every result of an application of the procedure R to X has the property A, then Y has property A. Proposition 77 of [Frege1879] p. 62. (Contributed by RP, 29-Jun-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- R e. W   &    |- A e. B   =>    |- ( X ( t+ ` R ) Y -> ( R hereditary A -> ( A. a ( X R a -> a e. A ) -> Y e. A ) ) )
Theorem	frege78 38235*	Commuted form of of frege77 38234. Proposition 78 of [Frege1879] p. 63. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- R e. W   &    |- A e. B   =>    |- ( R hereditary A -> ( A. a ( X R a -> a e. A ) -> ( X ( t+ ` R ) Y -> Y e. A ) ) )
Theorem	frege79 38236*	Distributed form of frege78 38235. Proposition 79 of [Frege1879] p. 63. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- R e. W   &    |- A e. B   =>    |- ( ( R hereditary A -> A. a ( X R a -> a e. A ) ) -> ( R hereditary A -> ( X ( t+ ` R ) Y -> Y e. A ) ) )
Theorem	frege80 38237*	Add additional condition to both clauses of frege79 38236. Proposition 80 of [Frege1879] p. 63. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- R e. W   &    |- A e. B   =>    |- ( ( X e. A -> ( R hereditary A -> A. a ( X R a -> a e. A ) ) ) -> ( X e. A -> ( R hereditary A -> ( X ( t+ ` R ) Y -> Y e. A ) ) ) )
Theorem	frege81 38238	If X has a property A that is hereditary in the R -sequence, and if Y follows X in the R-sequence, then Y has property A. This is a form of induction attributed to Jakob Bernoulli. Proposition 81 of [Frege1879] p. 63. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- R e. W   &    |- A e. B   =>    |- ( X e. A -> ( R hereditary A -> ( X ( t+ ` R ) Y -> Y e. A ) ) )
Theorem	frege82 38239	Closed-form deduction based on frege81 38238. Proposition 82 of [Frege1879] p. 64. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- R e. W   &    |- A e. B   =>    |- ( ( ph -> X e. A ) -> ( R hereditary A -> ( ph -> ( X ( t+ ` R ) Y -> Y e. A ) ) ) )
Theorem	frege83 38240	Apply commuted form of frege81 38238 when the property R is hereditary in a disjunction of two properties, only one of which is known to be held by X. Proposition 83 of [Frege1879] p. 65. Here we introduce the union of classes where Frege has a disjunction of properties which are represented by membership in either of the classes. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
|- X e. S   &    |- Y e. T   &    |- R e. U   &    |- B e. V   &    |- C e. W   =>    |- ( R hereditary ( B u. C ) -> ( X e. B -> ( X ( t+ ` R ) Y -> Y e. ( B u. C ) ) ) )
Theorem	frege84 38241	Commuted form of frege81 38238. Proposition 84 of [Frege1879] p. 65. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- R e. W   &    |- A e. B   =>    |- ( R hereditary A -> ( X e. A -> ( X ( t+ ` R ) Y -> Y e. A ) ) )
Theorem	frege85 38242*	Commuted form of frege77 38234. Proposition 85 of [Frege1879] p. 66. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- R e. W   &    |- A e. B   =>    |- ( X ( t+ ` R ) Y -> ( A. z ( X R z -> z e. A ) -> ( R hereditary A -> Y e. A ) ) )
Theorem	frege86 38243*	Conclusion about element one past Y in the R-sequence. Proposition 86 of [Frege1879] p. 66. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- R e. W   &    |- A e. B   =>    |- ( ( ( R hereditary A -> Y e. A ) -> ( R hereditary A -> ( Y R Z -> Z e. A ) ) ) -> ( X ( t+ ` R ) Y -> ( A. w ( X R w -> w e. A ) -> ( R hereditary A -> ( Y R Z -> Z e. A ) ) ) ) )
Theorem	frege87 38244*	If Z is a result of an application of the procedure R to an object Y that follows X in the R-sequence and if every result of an application of the procedure R to X has a property A that is hereditary in the R-sequence, then Z has property A. Proposition 87 of [Frege1879] p. 66. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- Z e. W   &    |- R e. S   &    |- A e. B   =>    |- ( X ( t+ ` R ) Y -> ( A. w ( X R w -> w e. A ) -> ( R hereditary A -> ( Y R Z -> Z e. A ) ) ) )
Theorem	frege88 38245*	Commuted form of frege87 38244. Proposition 88 of [Frege1879] p. 67. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- Z e. W   &    |- R e. S   &    |- A e. B   =>    |- ( Y R Z -> ( X ( t+ ` R ) Y -> ( A. w ( X R w -> w e. A ) -> ( R hereditary A -> Z e. A ) ) ) )
Theorem	frege89 38246*	One direction of dffrege76 38233. Proposition 89 of [Frege1879] p. 68. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- R e. W   =>    |- ( A. f ( R hereditary f -> ( A. w ( X R w -> w e. f ) -> Y e. f ) ) -> X ( t+ ` R ) Y )
Theorem	frege90 38247*	Add antecedent to frege89 38246. Proposition 90 of [Frege1879] p. 68. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- R e. W   =>    |- ( ( ph -> A. f ( R hereditary f -> ( A. w ( X R w -> w e. f ) -> Y e. f ) ) ) -> ( ph -> X ( t+ ` R ) Y ) )
Theorem	frege91 38248	Every result of an application of a procedure R to an object X follows that X in the R-sequence. Proposition 91 of [Frege1879] p. 68. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- R e. W   =>    |- ( X R Y -> X ( t+ ` R ) Y )
Theorem	frege92 38249	Inference from frege91 38248. Proposition 92 of [Frege1879] p. 69. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- R e. W   =>    |- ( X = Z -> ( X R Y -> Z ( t+ ` R ) Y ) )
Theorem	frege93 38250*	Necessary condition for two elements to be related by the transitive closure. Proposition 93 of [Frege1879] p. 70. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- R e. W   =>    |- ( A. f ( A. z ( X R z -> z e. f ) -> ( R hereditary f -> Y e. f ) ) -> X ( t+ ` R ) Y )
Theorem	frege94 38251*	Looking one past a pair related by transitive closure of a relation. Proposition 94 of [Frege1879] p. 70. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Z e. V   &    |- R e. W   =>    |- ( ( Y R Z -> ( X ( t+ ` R ) Y -> A. f ( A. w ( X R w -> w e. f ) -> ( R hereditary f -> Z e. f ) ) ) ) -> ( Y R Z -> ( X ( t+ ` R ) Y -> X ( t+ ` R ) Z ) ) )
Theorem	frege95 38252	Looking one past a pair related by transitive closure of a relation. Proposition 95 of [Frege1879] p. 70. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- Z e. W   &    |- R e. A   =>    |- ( Y R Z -> ( X ( t+ ` R ) Y -> X ( t+ ` R ) Z ) )
Theorem	frege96 38253	Every result of an application of the procedure R to an object that follows X in the R-sequence follows X in the R -sequence. Proposition 96 of [Frege1879] p. 71. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- Z e. W   &    |- R e. A   =>    |- ( X ( t+ ` R ) Y -> ( Y R Z -> X ( t+ ` R ) Z ) )
Theorem	frege97 38254	The property of following X in the R-sequence is hereditary in the R-sequence. Proposition 97 of [Frege1879] p. 71. Here we introduce the image of a singleton under a relation as class which stands for the property of following X in the R -sequence. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- R e. W   =>    |- R hereditary ( ( t+ ` R ) " { X } )
Theorem	frege98 38255	If Y follows X and Z follows Y in the R-sequence then Z follows X in the R-sequence because the transitive closure of a relation has the transitive property. Proposition 98 of [Frege1879] p. 71. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 6-Jul-2020.) (Proof modification is discouraged.)
|- X e. A   &    |- Y e. B   &    |- Z e. C   &    |- R e. D   =>    |- ( X ( t+ ` R ) Y -> ( Y ( t+ ` R ) Z -> X ( t+ ` R ) Z ) )
20.27.3.10  _Begriffsschrift_ Chapter III Member of sequence p ( ( t+ ` R ) u. _I ) c means c is a member of the R -sequence begining with p and p is a member of the R -sequence ending with c. dffrege99 38256 through frege114 38271 develop this. This will be shown to be related to the transitive-reflexive closure of relation R. But more work needs to be done on transitive closure of relations before this is ready for Metamath.
Theorem	dffrege99 38256	If Z is identical with X or follows X in the R -sequence, then we say : " Z belongs to the R-sequence beginning with X " or " X belongs to the R-sequence ending with Z". Definition 99 of [Frege1879] p. 71. (Contributed by RP, 2-Jul-2020.)
|- Z e. U   =>    |- ( ( -. X ( t+ ` R ) Z -> Z = X ) <-> X ( ( t+ ` R ) u. _I ) Z )
Theorem	frege100 38257	One direction of dffrege99 38256. Proposition 100 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
|- Z e. U   =>    |- ( X ( ( t+ ` R ) u. _I ) Z -> ( -. X ( t+ ` R ) Z -> Z = X ) )
Theorem	frege101 38258	Lemma for frege102 38259. Proposition 101 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
|- Z e. U   =>    |- ( ( Z = X -> ( Z R V -> X ( t+ ` R ) V ) ) -> ( ( X ( t+ ` R ) Z -> ( Z R V -> X ( t+ ` R ) V ) ) -> ( X ( ( t+ ` R ) u. _I ) Z -> ( Z R V -> X ( t+ ` R ) V ) ) ) )
Theorem	frege102 38259	If Z belongs to the R-sequence beginning with X, then every result of an application of the procedure R to Z follows X in the R-sequence. Proposition 102 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
|- X e. A   &    |- Z e. B   &    |- V e. C   &    |- R e. D   =>    |- ( X ( ( t+ ` R ) u. _I ) Z -> ( Z R V -> X ( t+ ` R ) V ) )
Theorem	frege103 38260	Proposition 103 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
|- Z e. V   =>    |- ( ( Z = X -> X = Z ) -> ( X ( ( t+ ` R ) u. _I ) Z -> ( -. X ( t+ ` R ) Z -> X = Z ) ) )
Theorem	frege104 38261	Proposition 104 of [Frege1879] p. 73. Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the minor clause and result swapped. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
|- Z e. V   =>    |- ( X ( ( t+ ` R ) u. _I ) Z -> ( -. X ( t+ ` R ) Z -> X = Z ) )
Theorem	frege105 38262	Proposition 105 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
|- Z e. V   =>    |- ( ( -. X ( t+ ` R ) Z -> Z = X ) -> X ( ( t+ ` R ) u. _I ) Z )
Theorem	frege106 38263	Whatever follows X in the R-sequence belongs to the R -sequence beginning with X. Proposition 106 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
|- Z e. V   =>    |- ( X ( t+ ` R ) Z -> X ( ( t+ ` R ) u. _I ) Z )
Theorem	frege107 38264	Proposition 107 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
|- V e. A   =>    |- ( ( Z ( ( t+ ` R ) u. _I ) Y -> ( Y R V -> Z ( t+ ` R ) V ) ) -> ( Z ( ( t+ ` R ) u. _I ) Y -> ( Y R V -> Z ( ( t+ ` R ) u. _I ) V ) ) )
Theorem	frege108 38265	If Y belongs to the R-sequence beginning with Z, then every result of an application of the procedure R to Y belongs to the R-sequence beginning with Z. Proposition 108 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
|- Z e. A   &    |- Y e. B   &    |- V e. C   &    |- R e. D   =>    |- ( Z ( ( t+ ` R ) u. _I ) Y -> ( Y R V -> Z ( ( t+ ` R ) u. _I ) V ) )
Theorem	frege109 38266	The property of belonging to the R-sequence beginning with X is hereditary in the R-sequence. Proposition 109 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- R e. V   =>    |- R hereditary ( ( ( t+ ` R ) u. _I ) " { X } )
Theorem	frege110 38267*	Proposition 110 of [Frege1879] p. 75. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
|- X e. A   &    |- Y e. B   &    |- M e. C   &    |- R e. D   =>    |- ( A. a ( Y R a -> X ( ( t+ ` R ) u. _I ) a ) -> ( Y ( t+ ` R ) M -> X ( ( t+ ` R ) u. _I ) M ) )
Theorem	frege111 38268	If Y belongs to the R-sequence beginning with Z, then every result of an application of the procedure R to Y belongs to the R-sequence beginning with Z or precedes Z in the R-sequence. Proposition 111 of [Frege1879] p. 75. (Contributed by RP, 7-Jul-2020.) (Revised by RP, 8-Jul-2020.) (Proof modification is discouraged.)
|- Z e. A   &    |- Y e. B   &    |- V e. C   &    |- R e. D   =>    |- ( Z ( ( t+ ` R ) u. _I ) Y -> ( Y R V -> ( -. V ( t+ ` R ) Z -> Z ( ( t+ ` R ) u. _I ) V ) ) )
Theorem	frege112 38269	Identity implies belonging to the R-sequence beginning with self. Proposition 112 of [Frege1879] p. 76. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
|- Z e. V   =>    |- ( Z = X -> X ( ( t+ ` R ) u. _I ) Z )
Theorem	frege113 38270	Proposition 113 of [Frege1879] p. 76. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
|- Z e. V   =>    |- ( ( Z ( ( t+ ` R ) u. _I ) X -> ( -. Z ( t+ ` R ) X -> Z = X ) ) -> ( Z ( ( t+ ` R ) u. _I ) X -> ( -. Z ( t+ ` R ) X -> X ( ( t+ ` R ) u. _I ) Z ) ) )
Theorem	frege114 38271	If X belongs to the R-sequence beginning with Z, then Z belongs to the R-sequence beginning with X or X follows Z in the R-sequence. Proposition 114 of [Frege1879] p. 76. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Z e. V   =>    |- ( Z ( ( t+ ` R ) u. _I ) X -> ( -. Z ( t+ ` R ) X -> X ( ( t+ ` R ) u. _I ) Z ) )
20.27.3.11  _Begriffsschrift_ Chapter III Single-valued procedures Fun `' `' R means the relationship content of procedure R is single-valued. The double converse allows us to simply apply this syntax in place of Frege's even though the original never explicitly limited discussion of propositional statments which vary on two variables to relations. dffrege115 38272 through frege133 38290 develop this and how functions relate to transitive and transitive-reflexive closures.
Theorem	dffrege115 38272*	If from the the circumstance that c is a result of an application of the procedure R to b, whatever b may be, it can be inferred that every result of an application of the procedure R to b is the same as c, then we say : "The procedure R is single-valued". Definition 115 of [Frege1879] p. 77. (Contributed by RP, 7-Jul-2020.)
|- ( A. c A. b ( b R c -> A. a ( b R a -> a = c ) ) <-> Fun `' `' R )
Theorem	frege116 38273*	One direction of dffrege115 38272. Proposition 116 of [Frege1879] p. 77. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   =>    |- ( Fun `' `' R -> A. b ( b R X -> A. a ( b R a -> a = X ) ) )
Theorem	frege117 38274*	Lemma for frege118 38275. Proposition 117 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   =>    |- ( ( A. b ( b R X -> A. a ( b R a -> a = X ) ) -> ( Y R X -> A. a ( Y R a -> a = X ) ) ) -> ( Fun `' `' R -> ( Y R X -> A. a ( Y R a -> a = X ) ) ) )
Theorem	frege118 38275*	Simplified application of one direction of dffrege115 38272. Proposition 118 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   =>    |- ( Fun `' `' R -> ( Y R X -> A. a ( Y R a -> a = X ) ) )
Theorem	frege119 38276*	Lemma for frege120 38277. Proposition 119 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   =>    |- ( ( A. a ( Y R a -> a = X ) -> ( Y R A -> A = X ) ) -> ( Fun `' `' R -> ( Y R X -> ( Y R A -> A = X ) ) ) )
Theorem	frege120 38277	Simplified application of one direction of dffrege115 38272. Proposition 120 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- A e. W   =>    |- ( Fun `' `' R -> ( Y R X -> ( Y R A -> A = X ) ) )
Theorem	frege121 38278	Lemma for frege122 38279. Proposition 121 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- A e. W   =>    |- ( ( A = X -> X ( ( t+ ` R ) u. _I ) A ) -> ( Fun `' `' R -> ( Y R X -> ( Y R A -> X ( ( t+ ` R ) u. _I ) A ) ) ) )
Theorem	frege122 38279	If X is a result of an application of the single-valued procedure R to Y, then every result of an application of the procedure R to Y belongs to the R-sequence beginning with X. Proposition 122 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- A e. W   =>    |- ( Fun `' `' R -> ( Y R X -> ( Y R A -> X ( ( t+ ` R ) u. _I ) A ) ) )
Theorem	frege123 38280*	Lemma for frege124 38281. Proposition 123 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   =>    |- ( ( A. a ( Y R a -> X ( ( t+ ` R ) u. _I ) a ) -> ( Y ( t+ ` R ) M -> X ( ( t+ ` R ) u. _I ) M ) ) -> ( Fun `' `' R -> ( Y R X -> ( Y ( t+ ` R ) M -> X ( ( t+ ` R ) u. _I ) M ) ) ) )
Theorem	frege124 38281	If X is a result of an application of the single-valued procedure R to Y and if M follows Y in the R-sequence, then M belongs to the R-sequence beginning with X. Proposition 124 of [Frege1879] p. 80. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- M e. W   &    |- R e. S   =>    |- ( Fun `' `' R -> ( Y R X -> ( Y ( t+ ` R ) M -> X ( ( t+ ` R ) u. _I ) M ) ) )
Theorem	frege125 38282	Lemma for frege126 38283. Proposition 125 of [Frege1879] p. 81. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- M e. W   &    |- R e. S   =>    |- ( ( X ( ( t+ ` R ) u. _I ) M -> ( -. X ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) X ) ) -> ( Fun `' `' R -> ( Y R X -> ( Y ( t+ ` R ) M -> ( -. X ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) X ) ) ) ) )
Theorem	frege126 38283	If M follows Y in the R-sequence and if the procedure R is single-valued, then every result of an application of the procedure R to Y belongs to the R-sequence beginning with M or precedes M in the R-sequence. Proposition 126 of [Frege1879] p. 81. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- M e. W   &    |- R e. S   =>    |- ( Fun `' `' R -> ( Y R X -> ( Y ( t+ ` R ) M -> ( -. X ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) X ) ) ) )
Theorem	frege127 38284	Communte antecedents of frege126 38283. Proposition 127 of [Frege1879] p. 82. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- M e. W   &    |- R e. S   =>    |- ( Fun `' `' R -> ( Y ( t+ ` R ) M -> ( Y R X -> ( -. X ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) X ) ) ) )
Theorem	frege128 38285	Lemma for frege129 38286. Proposition 128 of [Frege1879] p. 83. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- M e. W   &    |- R e. S   =>    |- ( ( M ( ( t+ ` R ) u. _I ) Y -> ( Y R X -> ( -. X ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) X ) ) ) -> ( Fun `' `' R -> ( ( -. Y ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) Y ) -> ( Y R X -> ( -. X ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) X ) ) ) ) )
Theorem	frege129 38286	If the procedure R is single-valued and Y belongs to the R -sequence begining with M or precedes M in the R-sequence, then every result of an application of the procedure R to Y belongs to the R-sequence begining with M or precedes M in the R-sequence. Proposition 129 of [Frege1879] p. 83. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- M e. W   &    |- R e. S   =>    |- ( Fun `' `' R -> ( ( -. Y ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) Y ) -> ( Y R X -> ( -. X ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) X ) ) ) )
Theorem	frege130 38287*	Lemma for frege131 38288. Proposition 130 of [Frege1879] p. 84. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
|- M e. U   &    |- R e. V   =>    |- ( ( A. b ( ( -. b ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) b ) -> A. a ( b R a -> ( -. a ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) a ) ) ) -> R hereditary ( ( `' ( t+ ` R ) " { M } ) u. ( ( ( t+ ` R ) u. _I ) " { M } ) ) ) -> ( Fun `' `' R -> R hereditary ( ( `' ( t+ ` R ) " { M } ) u. ( ( ( t+ ` R ) u. _I ) " { M } ) ) ) )
Theorem	frege131 38288	If the procedure R is single-valued, then the property of belonging to the R-sequence begining with M or preceeding M in the R-sequence is hereditary in the R-sequence. Proposition 131 of [Frege1879] p. 85. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
|- M e. U   &    |- R e. V   =>    |- ( Fun `' `' R -> R hereditary ( ( `' ( t+ ` R ) " { M } ) u. ( ( ( t+ ` R ) u. _I ) " { M } ) ) )
Theorem	frege132 38289	Lemma for frege133 38290. Proposition 132 of [Frege1879] p. 86. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
|- M e. U   &    |- R e. V   =>    |- ( ( R hereditary ( ( `' ( t+ ` R ) " { M } ) u. ( ( ( t+ ` R ) u. _I ) " { M } ) ) -> ( X ( t+ ` R ) M -> ( X ( t+ ` R ) Y -> ( -. Y ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) Y ) ) ) ) -> ( Fun `' `' R -> ( X ( t+ ` R ) M -> ( X ( t+ ` R ) Y -> ( -. Y ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) Y ) ) ) ) )
Theorem	frege133 38290	If the procedure R is single-valued and if M and Y follow X in the R-sequence, then Y belongs to the R-sequence beginning with M or precedes M in the R-sequence. Proposition 133 of [Frege1879] p. 86. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
|- X e. U   &    |- Y e. V   &    |- M e. W   &    |- R e. S   =>    |- ( Fun `' `' R -> ( X ( t+ ` R ) M -> ( X ( t+ ` R ) Y -> ( -. Y ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) Y ) ) ) )
20.27.4  Exploring Topology via Seifert And Threlfall See Seifert And Threlfall: A Textbook Of Topology (1980) which is an English translation of Lehrbuch der Topologie (1934).
20.27.4.1  Equinumerosity of sets of relations and maps Because ( ( 2o ^m B ) ^m A ) ~~ ( 2o ^m ( A X. B ) ) ~~ ( ( 2o ^m A ) ^m B ) is an instance of the law of exponents: ( ( C ^m B ) ^m A ) ~~ ( C ^m ( A X. B ) ) ~~ ( ( C ^m A ) ^m B ) we are led to see that ( ~P B ^m A ) ~~ ~P ( A X. B ) ~~ ( ~P A ^m B ) is true for any two sets, A and B, and thus there exist one-to-one onto relations between each of these three sets of relations.
Theorem	enrelmap 38291	The set of all possible relations between two sets is equinumerous to the set of all mappings from one set to the powerset of the other. See rfovf1od 38300 for a demonstration of an natural one-to-one onto mapping. (Contributed by RP, 27-Apr-2021.)
|- ( ( A e. V /\ B e. W ) -> ~P ( A X. B ) ~~ ( ~P B ^m A ) )
Theorem	enrelmapr 38292	The set of all possible relations between two sets is equinumerous to the set of all mappings from one set to the powerset of the other. (Contributed by RP, 27-Apr-2021.)
|- ( ( A e. V /\ B e. W ) -> ~P ( A X. B ) ~~ ( ~P A ^m B ) )
Theorem	enmappw 38293	The set of all mappings from one set to the powerset of the other is equinumerous to the set of all mappings from the second set to the powerset of the first. (Contributed by RP, 27-Apr-2021.)
|- ( ( A e. V /\ B e. W ) -> ( ~P B ^m A ) ~~ ( ~P A ^m B ) )
Theorem	enmappwid 38294	The set of all mappings from the powerset to the powerset is equinumerous to the set of all mappings from the set to the powerset of the powerset. (Contributed by RP, 27-Apr-2021.)
|- ( A e. V -> ( ~P A ^m ~P A ) ~~ ( ~P ~P A ^m A ) )
Theorem	rfovd 38295*	Value of the operator, ( A O B ), which maps between relations and functions for relations between base sets, A and B. (Contributed by RP, 25-Apr-2021.)
|- O = ( a e. _V , b e. _V |-> ( r e. ~P ( a X. b ) |-> ( x e. a |-> { y e. b | x r y } ) ) )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   =>    |- ( ph -> ( A O B ) = ( r e. ~P ( A X. B ) |-> ( x e. A |-> { y e. B | x r y } ) ) )
Theorem	rfovfvd 38296*	Value of the operator, ( A O B ), which maps between relations and functions for relations between base sets, A and B, and relation R. (Contributed by RP, 25-Apr-2021.)
|- O = ( a e. _V , b e. _V |-> ( r e. ~P ( a X. b ) |-> ( x e. a |-> { y e. b | x r y } ) ) )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ph -> R e. ~P ( A X. B ) )   &    |- F = ( A O B )   =>    |- ( ph -> ( F ` R ) = ( x e. A |-> { y e. B | x R y } ) )
Theorem	rfovfvfvd 38297*	Value of the operator, ( A O B ), which maps between relations and functions for relations between base sets, A and B, relation R, and left element X. (Contributed by RP, 25-Apr-2021.)
|- O = ( a e. _V , b e. _V |-> ( r e. ~P ( a X. b ) |-> ( x e. a |-> { y e. b | x r y } ) ) )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ph -> R e. ~P ( A X. B ) )   &    |- F = ( A O B )   &    |- ( ph -> X e. A )   &    |- G = ( F ` R )   =>    |- ( ph -> ( G ` X ) = { y e. B | X R y } )
Theorem	rfovcnvf1od 38298*	Properties of the operator, ( A O B ), which maps between relations and functions for relations between base sets, A and B. (Contributed by RP, 27-Apr-2021.)
|- O = ( a e. _V , b e. _V |-> ( r e. ~P ( a X. b ) |-> ( x e. a |-> { y e. b | x r y } ) ) )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- F = ( A O B )   =>    |- ( ph -> ( F : ~P ( A X. B ) -1-1-onto-> ( ~P B ^m A ) /\ `' F = ( f e. ( ~P B ^m A ) |-> { <. x , y >. | ( x e. A /\ y e. ( f ` x ) ) } ) ) )
Theorem	rfovcnvd 38299*	Value of the converse of the operator, ( A O B ), which maps between relations and functions for relations between base sets, A and B. (Contributed by RP, 27-Apr-2021.)
|- O = ( a e. _V , b e. _V |-> ( r e. ~P ( a X. b ) |-> ( x e. a |-> { y e. b | x r y } ) ) )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- F = ( A O B )   =>    |- ( ph -> `' F = ( f e. ( ~P B ^m A ) |-> { <. x , y >. | ( x e. A /\ y e. ( f ` x ) ) } ) )
Theorem	rfovf1od 38300*	The value of the operator, ( A O B ), which maps between relations and functions for relations between base sets, A and B, is a bijection. (Contributed by RP, 27-Apr-2021.)
|- O = ( a e. _V , b e. _V |-> ( r e. ~P ( a X. b ) |-> ( x e. a |-> { y e. b | x r y } ) ) )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- F = ( A O B )   =>    |- ( ph -> F : ~P ( A X. B ) -1-1-onto-> ( ~P B ^m A ) )