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Theorem List for Metamath Proof Explorer - 37201-37300   *Has distinct variable group(s)
TypeLabelDescription
Statement

21.26.3.6  _Begriffsschrift_ Chapter II with equivalence of sets

Theoremaxfrege52c 37201 Justification for ax-frege52c 37202. (Contributed by RP, 24-Dec-2019.)
(𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑))

Axiomax-frege52c 37202 One side of dfsbcq 3404. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
(𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑))

Theoremfrege52b 37203 The case when the content of 𝑥 is identical with the content of 𝑦 and in which a proposition controlled by an element for which we substitute the content of 𝑥 is affirmed and the same proposition, this time where we subsitute the content of 𝑦, is denied does not take place. In [𝑥 / 𝑧]𝜑, 𝑥 can also occur in other than the argument (𝑧) places. Hence 𝑥 may still be contained in [𝑦 / 𝑧]𝜑. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))

Theoremfrege53b 37204 Lemma for frege102 (via frege92 37269). Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
([𝑥 / 𝑦]𝜑 → (𝑥 = 𝑧 → [𝑧 / 𝑦]𝜑))

Theoremaxfrege54c 37205 Reflexive equality of classes. Identical to eqid 2610. Justification for ax-frege54c 37206. (Contributed by RP, 24-Dec-2019.)
𝐴 = 𝐴

Axiomax-frege54c 37206 Reflexive equality of sets (as classes). Part of Axiom 54 of [Frege1879] p. 50. Identical to eqid 2610. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
𝐴 = 𝐴

Theoremfrege54b 37207 Reflexive equality of sets. The content of 𝑥 is identical with the content of 𝑥. Part of Axiom 54 of [Frege1879] p. 50. Slightly specialized version of eqid 2610. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝑥 = 𝑥

Theoremfrege54cor1b 37208 Reflexive equality. (Contributed by RP, 24-Dec-2019.)
[𝑥 / 𝑦]𝑦 = 𝑥

Theoremfrege55lem1b 37209* Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.)
((𝜑 → [𝑥 / 𝑦]𝑦 = 𝑧) → (𝜑𝑥 = 𝑧))

Theoremfrege55lem2b 37210 Lemma for frege55b 37211. Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥)

Theoremfrege55b 37211 Lemma for frege57b 37213. Proposition 55 of [Frege1879] p. 50.

Note that eqtr2 2630 incorporates eqcom 2617 which is stronger than this proposition which is identical to equcomi 1931. Is is possible that Frege tricked himself into assuming what he was out to prove? (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

(𝑥 = 𝑦𝑦 = 𝑥)

Theoremfrege56b 37212 Lemma for frege57b 37213. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) → (𝑦 = 𝑥 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))

Theoremfrege57b 37213 Analogue of frege57aid 37186. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 → [𝑥 / 𝑧]𝜑))

Theoremaxfrege58b 37214 If 𝑥𝜑 is affirmed, [𝑦 / 𝑥]𝜑 cannot be denied. Identical to stdpc4 2341. Justification for ax-frege58b 37215. (Contributed by RP, 28-Mar-2020.)
(∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)

Axiomax-frege58b 37215 If 𝑥𝜑 is affirmed, [𝑦 / 𝑥]𝜑 cannot be denied. Identical to stdpc4 2341. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (New usage is discouraged.)
(∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)

Theoremfrege58bid 37216 If 𝑥𝜑 is affirmed, 𝜑 cannot be denied. Identical to sp 2041. See ax-frege58b 37215 and frege58c 37235 for versions which more closely track the original. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (Proof modification is discouraged.)
(∀𝑥𝜑𝜑)

Theoremfrege58bcor 37217 Lemma for frege59b 37218. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Theoremfrege59b 37218 A kind of Aristotelian inference. Namely Felapton or Fesapo. 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 37127 incorrectly referenced where frege30 37146 is in the original. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

([𝑥 / 𝑦]𝜑 → (¬ [𝑥 / 𝑦]𝜓 → ¬ ∀𝑦(𝜑𝜓)))

Theoremfrege60b 37219 Swap antecedents of ax-frege58b 37215. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(∀𝑥(𝜑 → (𝜓𝜒)) → ([𝑦 / 𝑥]𝜓 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))

Theoremfrege61b 37220 Lemma for frege65b 37224. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(([𝑥 / 𝑦]𝜑𝜓) → (∀𝑦𝜑𝜓))

Theoremfrege62b 37221 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2551 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]𝜑 → (∀𝑦(𝜑𝜓) → [𝑥 / 𝑦]𝜓))

Theoremfrege63b 37222 Lemma for frege91 37268. Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
([𝑥 / 𝑦]𝜑 → (𝜓 → (∀𝑦(𝜑𝜒) → [𝑥 / 𝑦]𝜒)))

Theoremfrege64b 37223 Lemma for frege65b 37224. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜓) → (∀𝑦(𝜓𝜒) → ([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜒)))

Theoremfrege65b 37224 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2551 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 : (∀𝑥([𝑥 / 𝑎]𝜑 → [𝑥 / 𝑏]𝜓) → (∀𝑦([𝑦 / 𝑏]𝜓 → [𝑦 / 𝑐]𝜒) → ([𝑧 / 𝑎]𝜑 → [𝑧 / 𝑐]𝜒))). But that is perhaps too pedantic a translation for this exploration. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

(∀𝑥(𝜑𝜓) → (∀𝑥(𝜓𝜒) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))

Theoremfrege66b 37225 Swap antecedents of frege65b 37224. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(∀𝑥(𝜑𝜓) → (∀𝑥(𝜒𝜑) → ([𝑦 / 𝑥]𝜒 → [𝑦 / 𝑥]𝜓)))

Theoremfrege67b 37226 Lemma for frege68b 37227. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((∀𝑥𝜑𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑𝜓) → (𝜓 → [𝑦 / 𝑥]𝜑)))

Theoremfrege68b 37227 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.)
((∀𝑥𝜑𝜓) → (𝜓 → [𝑦 / 𝑥]𝜑))

21.26.3.7  _Begriffsschrift_ Chapter II with equivalence of classes (where they are sets)

Theoremfrege53c 37228 Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
([𝐴 / 𝑥]𝜑 → (𝐴 = 𝐵[𝐵 / 𝑥]𝜑))

Theoremfrege54cor1c 37229* Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Revised by RP, 25-Apr-2020.)
𝐴𝐶       [𝐴 / 𝑥]𝑥 = 𝐴

Theoremfrege55lem1c 37230* Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.)
((𝜑[𝐴 / 𝑥]𝑥 = 𝐵) → (𝜑𝐴 = 𝐵))

Theoremfrege55lem2c 37231* 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)

Theoremfrege55c 37232 Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝑥 = 𝐴𝐴 = 𝑥)

Theoremfrege56c 37233* Lemma for frege57c 37234. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝐵𝐶       ((𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑)) → (𝐵 = 𝐴 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑)))

Theoremfrege57c 37234* Swap order of implication in ax-frege52c 37202. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝐴𝐶       (𝐴 = 𝐵 → ([𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))

Theoremfrege58c 37235 Principle related to sp 2041. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝐴𝐵       (∀𝑥𝜑[𝐴 / 𝑥]𝜑)

Theoremfrege59c 37236 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 37127 incorrectly referenced where frege30 37146 is in the original. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

𝐴𝐵       ([𝐴 / 𝑥]𝜑 → (¬ [𝐴 / 𝑥]𝜓 → ¬ ∀𝑥(𝜑𝜓)))

Theoremfrege60c 37237 Swap antecedents of frege58c 37235. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝐴𝐵       (∀𝑥(𝜑 → (𝜓𝜒)) → ([𝐴 / 𝑥]𝜓 → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜒)))

Theoremfrege61c 37238 Lemma for frege65c 37242. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝐴𝐵       (([𝐴 / 𝑥]𝜑𝜓) → (∀𝑥𝜑𝜓))

Theoremfrege62c 37239 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2551 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝐴𝐵       ([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑𝜓) → [𝐴 / 𝑥]𝜓))

Theoremfrege63c 37240 Analogue of frege63b 37222. Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝐴𝐵       ([𝐴 / 𝑥]𝜑 → (𝜓 → (∀𝑥(𝜑𝜒) → [𝐴 / 𝑥]𝜒)))

Theoremfrege64c 37241 Lemma for frege65c 37242. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝐴𝐵       (([𝐶 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) → (∀𝑥(𝜓𝜒) → ([𝐶 / 𝑥]𝜑[𝐴 / 𝑥]𝜒)))

Theoremfrege65c 37242 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2551 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝐴𝐵       (∀𝑥(𝜑𝜓) → (∀𝑥(𝜓𝜒) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜒)))

Theoremfrege66c 37243 Swap antecedents of frege65c 37242. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝐴𝐵       (∀𝑥(𝜑𝜓) → (∀𝑥(𝜒𝜑) → ([𝐴 / 𝑥]𝜒[𝐴 / 𝑥]𝜓)))

Theoremfrege67c 37244 Lemma for frege68c 37245. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝐴𝐵       (((∀𝑥𝜑𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑𝜓) → (𝜓[𝐴 / 𝑥]𝜑)))

Theoremfrege68c 37245 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.)
𝐴𝐵       ((∀𝑥𝜑𝜓) → (𝜓[𝐴 / 𝑥]𝜑))

21.26.3.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence

(𝑅𝐴) ⊆ 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 37274 and frege109 37286.

dffrege69 37246 through frege75 37252 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.

Theoremdffrege69 37246* 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.)
(∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ 𝑅 hereditary 𝐴)

Theoremfrege70 37247* Lemma for frege72 37249. Proposition 70 of [Frege1879] p. 58. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑉       (𝑅 hereditary 𝐴 → (𝑋𝐴 → ∀𝑦(𝑋𝑅𝑦𝑦𝐴)))

Theoremfrege71 37248* Lemma for frege72 37249. Proposition 71 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑉       ((∀𝑧(𝑋𝑅𝑧𝑧𝐴) → (𝑋𝑅𝑌𝑌𝐴)) → (𝑅 hereditary 𝐴 → (𝑋𝐴 → (𝑋𝑅𝑌𝑌𝐴))))

Theoremfrege72 37249 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.)
𝑋𝑈    &   𝑌𝑉       (𝑅 hereditary 𝐴 → (𝑋𝐴 → (𝑋𝑅𝑌𝑌𝐴)))

Theoremfrege73 37250 Lemma for frege87 37264. Proposition 73 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉       ((𝑅 hereditary 𝐴𝑋𝐴) → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌𝑌𝐴)))

Theoremfrege74 37251 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.)
𝑋𝑈    &   𝑌𝑉       (𝑋𝐴 → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌𝑌𝐴)))

Theoremfrege75 37252* 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.)
(∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) → 𝑅 hereditary 𝐴)

21.26.3.9  _Begriffsschrift_ Chapter III Following in a sequence

𝑝(t+‘𝑅)𝑐 means 𝑐 follows 𝑝 in the 𝑅-sequence.

dffrege76 37253 through frege98 37275 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.

Theoremdffrege76 37253* 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.)

𝐵𝑈    &   𝐸𝑉    &   𝑅𝑊       (∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝐵𝑅𝑎𝑎𝑓) → 𝐸𝑓)) ↔ 𝐵(t+‘𝑅)𝐸)

Theoremfrege77 37254* 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.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊    &   𝐴𝐵       (𝑋(t+‘𝑅)𝑌 → (𝑅 hereditary 𝐴 → (∀𝑎(𝑋𝑅𝑎𝑎𝐴) → 𝑌𝐴)))

Theoremfrege78 37255* Commuted form of of frege77 37254. Proposition 78 of [Frege1879] p. 63. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊    &   𝐴𝐵       (𝑅 hereditary 𝐴 → (∀𝑎(𝑋𝑅𝑎𝑎𝐴) → (𝑋(t+‘𝑅)𝑌𝑌𝐴)))

Theoremfrege79 37256* Distributed form of frege78 37255. Proposition 79 of [Frege1879] p. 63. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊    &   𝐴𝐵       ((𝑅 hereditary 𝐴 → ∀𝑎(𝑋𝑅𝑎𝑎𝐴)) → (𝑅 hereditary 𝐴 → (𝑋(t+‘𝑅)𝑌𝑌𝐴)))

Theoremfrege80 37257* Add additional condition to both clauses of frege79 37256. Proposition 80 of [Frege1879] p. 63. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊    &   𝐴𝐵       ((𝑋𝐴 → (𝑅 hereditary 𝐴 → ∀𝑎(𝑋𝑅𝑎𝑎𝐴))) → (𝑋𝐴 → (𝑅 hereditary 𝐴 → (𝑋(t+‘𝑅)𝑌𝑌𝐴))))

Theoremfrege81 37258 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.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊    &   𝐴𝐵       (𝑋𝐴 → (𝑅 hereditary 𝐴 → (𝑋(t+‘𝑅)𝑌𝑌𝐴)))

Theoremfrege82 37259 Closed-form deduction based on frege81 37258. Proposition 82 of [Frege1879] p. 64. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊    &   𝐴𝐵       ((𝜑𝑋𝐴) → (𝑅 hereditary 𝐴 → (𝜑 → (𝑋(t+‘𝑅)𝑌𝑌𝐴))))

Theoremfrege83 37260 Apply commuted form of frege81 37258 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.)
𝑋𝑆    &   𝑌𝑇    &   𝑅𝑈    &   𝐵𝑉    &   𝐶𝑊       (𝑅 hereditary (𝐵𝐶) → (𝑋𝐵 → (𝑋(t+‘𝑅)𝑌𝑌 ∈ (𝐵𝐶))))

Theoremfrege84 37261 Commuted form of frege81 37258. Proposition 84 of [Frege1879] p. 65. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊    &   𝐴𝐵       (𝑅 hereditary 𝐴 → (𝑋𝐴 → (𝑋(t+‘𝑅)𝑌𝑌𝐴)))

Theoremfrege85 37262* Commuted form of frege77 37254. Proposition 85 of [Frege1879] p. 66. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊    &   𝐴𝐵       (𝑋(t+‘𝑅)𝑌 → (∀𝑧(𝑋𝑅𝑧𝑧𝐴) → (𝑅 hereditary 𝐴𝑌𝐴)))

Theoremfrege86 37263* 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.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊    &   𝐴𝐵       (((𝑅 hereditary 𝐴𝑌𝐴) → (𝑅 hereditary 𝐴 → (𝑌𝑅𝑍𝑍𝐴))) → (𝑋(t+‘𝑅)𝑌 → (∀𝑤(𝑋𝑅𝑤𝑤𝐴) → (𝑅 hereditary 𝐴 → (𝑌𝑅𝑍𝑍𝐴)))))

Theoremfrege87 37264* 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.)
𝑋𝑈    &   𝑌𝑉    &   𝑍𝑊    &   𝑅𝑆    &   𝐴𝐵       (𝑋(t+‘𝑅)𝑌 → (∀𝑤(𝑋𝑅𝑤𝑤𝐴) → (𝑅 hereditary 𝐴 → (𝑌𝑅𝑍𝑍𝐴))))

Theoremfrege88 37265* Commuted form of frege87 37264. Proposition 88 of [Frege1879] p. 67. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑍𝑊    &   𝑅𝑆    &   𝐴𝐵       (𝑌𝑅𝑍 → (𝑋(t+‘𝑅)𝑌 → (∀𝑤(𝑋𝑅𝑤𝑤𝐴) → (𝑅 hereditary 𝐴𝑍𝐴))))

Theoremfrege89 37266* One direction of dffrege76 37253. Proposition 89 of [Frege1879] p. 68. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊       (∀𝑓(𝑅 hereditary 𝑓 → (∀𝑤(𝑋𝑅𝑤𝑤𝑓) → 𝑌𝑓)) → 𝑋(t+‘𝑅)𝑌)

Theoremfrege90 37267* Add antecedent to frege89 37266. Proposition 90 of [Frege1879] p. 68. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊       ((𝜑 → ∀𝑓(𝑅 hereditary 𝑓 → (∀𝑤(𝑋𝑅𝑤𝑤𝑓) → 𝑌𝑓))) → (𝜑𝑋(t+‘𝑅)𝑌))

Theoremfrege91 37268 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.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊       (𝑋𝑅𝑌𝑋(t+‘𝑅)𝑌)

Theoremfrege92 37269 Inference from frege91 37268. Proposition 92 of [Frege1879] p. 69. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊       (𝑋 = 𝑍 → (𝑋𝑅𝑌𝑍(t+‘𝑅)𝑌))

Theoremfrege93 37270* 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.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊       (∀𝑓(∀𝑧(𝑋𝑅𝑧𝑧𝑓) → (𝑅 hereditary 𝑓𝑌𝑓)) → 𝑋(t+‘𝑅)𝑌)

Theoremfrege94 37271* 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.)
𝑋𝑈    &   𝑍𝑉    &   𝑅𝑊       ((𝑌𝑅𝑍 → (𝑋(t+‘𝑅)𝑌 → ∀𝑓(∀𝑤(𝑋𝑅𝑤𝑤𝑓) → (𝑅 hereditary 𝑓𝑍𝑓)))) → (𝑌𝑅𝑍 → (𝑋(t+‘𝑅)𝑌𝑋(t+‘𝑅)𝑍)))

Theoremfrege95 37272 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.)
𝑋𝑈    &   𝑌𝑉    &   𝑍𝑊    &   𝑅𝐴       (𝑌𝑅𝑍 → (𝑋(t+‘𝑅)𝑌𝑋(t+‘𝑅)𝑍))

Theoremfrege96 37273 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.)
𝑋𝑈    &   𝑌𝑉    &   𝑍𝑊    &   𝑅𝐴       (𝑋(t+‘𝑅)𝑌 → (𝑌𝑅𝑍𝑋(t+‘𝑅)𝑍))

Theoremfrege97 37274 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.)

𝑋𝑈    &   𝑅𝑊       𝑅 hereditary ((t+‘𝑅) “ {𝑋})

Theoremfrege98 37275 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.)
𝑋𝐴    &   𝑌𝐵    &   𝑍𝐶    &   𝑅𝐷       (𝑋(t+‘𝑅)𝑌 → (𝑌(t+‘𝑅)𝑍𝑋(t+‘𝑅)𝑍))

21.26.3.10  _Begriffsschrift_ Chapter III Member of sequence

𝑝((t+‘𝑅) ∪ 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 37276 through frege114 37291 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.

Theoremdffrege99 37276 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.)
𝑍𝑈       ((¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍)

Theoremfrege100 37277 One direction of dffrege99 37276. Proposition 100 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑍𝑈       (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋))

Theoremfrege101 37278 Lemma for frege102 37279. Proposition 101 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑍𝑈       ((𝑍 = 𝑋 → (𝑍𝑅𝑉𝑋(t+‘𝑅)𝑉)) → ((𝑋(t+‘𝑅)𝑍 → (𝑍𝑅𝑉𝑋(t+‘𝑅)𝑉)) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (𝑍𝑅𝑉𝑋(t+‘𝑅)𝑉))))

Theoremfrege102 37279 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.)
𝑋𝐴    &   𝑍𝐵    &   𝑉𝐶    &   𝑅𝐷       (𝑋((t+‘𝑅) ∪ I )𝑍 → (𝑍𝑅𝑉𝑋(t+‘𝑅)𝑉))

Theoremfrege103 37280 Proposition 103 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑍𝑉       ((𝑍 = 𝑋𝑋 = 𝑍) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍𝑋 = 𝑍)))

Theoremfrege104 37281 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.)

𝑍𝑉       (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍𝑋 = 𝑍))

Theoremfrege105 37282 Proposition 105 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑍𝑉       ((¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋) → 𝑋((t+‘𝑅) ∪ I )𝑍)

Theoremfrege106 37283 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.)
𝑍𝑉       (𝑋(t+‘𝑅)𝑍𝑋((t+‘𝑅) ∪ I )𝑍)

Theoremfrege107 37284 Proposition 107 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑉𝐴       ((𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉𝑍(t+‘𝑅)𝑉)) → (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉𝑍((t+‘𝑅) ∪ I )𝑉)))

Theoremfrege108 37285 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.)
𝑍𝐴    &   𝑌𝐵    &   𝑉𝐶    &   𝑅𝐷       (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉𝑍((t+‘𝑅) ∪ I )𝑉))

Theoremfrege109 37286 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.)
𝑋𝑈    &   𝑅𝑉       𝑅 hereditary (((t+‘𝑅) ∪ I ) “ {𝑋})

Theoremfrege110 37287* Proposition 110 of [Frege1879] p. 75. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑋𝐴    &   𝑌𝐵    &   𝑀𝐶    &   𝑅𝐷       (∀𝑎(𝑌𝑅𝑎𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀𝑋((t+‘𝑅) ∪ I )𝑀))

Theoremfrege111 37288 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.)
𝑍𝐴    &   𝑌𝐵    &   𝑉𝐶    &   𝑅𝐷       (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → (¬ 𝑉(t+‘𝑅)𝑍𝑍((t+‘𝑅) ∪ I )𝑉)))

Theoremfrege112 37289 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.)
𝑍𝑉       (𝑍 = 𝑋𝑋((t+‘𝑅) ∪ I )𝑍)

Theoremfrege113 37290 Proposition 113 of [Frege1879] p. 76. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑍𝑉       ((𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋𝑍 = 𝑋)) → (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋𝑋((t+‘𝑅) ∪ I )𝑍)))

Theoremfrege114 37291 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.)
𝑋𝑈    &   𝑍𝑉       (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋𝑋((t+‘𝑅) ∪ I )𝑍))

21.26.3.11  _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 37292 through frege133 37310 develop this and how functions relate to transitive and transitive-reflexive closures.

Theoremdffrege115 37292* 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.)
(∀𝑐𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ Fun 𝑅)

Theoremfrege116 37293* One direction of dffrege115 37292. Proposition 116 of [Frege1879] p. 77. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈       (Fun 𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))

Theoremfrege117 37294* Lemma for frege118 37295. Proposition 117 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈       ((∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋))) → (Fun 𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋))))

Theoremfrege118 37295* Simplified application of one direction of dffrege115 37292. Proposition 118 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉       (Fun 𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋)))

Theoremfrege119 37296* Lemma for frege120 37297. Proposition 119 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉       ((∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋) → (𝑌𝑅𝐴𝐴 = 𝑋)) → (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴𝐴 = 𝑋))))

Theoremfrege120 37297 Simplified application of one direction of dffrege115 37292. Proposition 120 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝐴𝑊       (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴𝐴 = 𝑋)))

Theoremfrege121 37298 Lemma for frege122 37299. Proposition 121 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝐴𝑊       ((𝐴 = 𝑋𝑋((t+‘𝑅) ∪ I )𝐴) → (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴𝑋((t+‘𝑅) ∪ I )𝐴))))

Theoremfrege122 37299 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.)
𝑋𝑈    &   𝑌𝑉    &   𝐴𝑊       (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴𝑋((t+‘𝑅) ∪ I )𝐴)))

Theoremfrege123 37300* Lemma for frege124 37301. Proposition 123 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉       ((∀𝑎(𝑌𝑅𝑎𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀𝑋((t+‘𝑅) ∪ I )𝑀)) → (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀𝑋((t+‘𝑅) ∪ I )𝑀))))

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