P. 440 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43901-44000   *Has distinct variable group(s)

Type	Label	Description
Statement
Axiom	ax-frege52c 43901	One side of dfsbcq 3790. 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))
Theorem	frege52b 43902	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 substitute 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))
Theorem	frege53b 43903	Lemma for frege102 (via frege92 43968). Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ([𝑦 / 𝑥]𝜑 → (𝑦 = 𝑧 → [𝑧 / 𝑥]𝜑))
Theorem	axfrege54c 43904	Reflexive equality of classes. Identical to eqid 2737. Justification for ax-frege54c 43905. (Contributed by RP, 24-Dec-2019.)
⊢ 𝐴 = 𝐴
Axiom	ax-frege54c 43905	Reflexive equality of sets (as classes). Part of Axiom 54 of [Frege1879] p. 50. Identical to eqid 2737. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
⊢ 𝐴 = 𝐴
Theorem	frege54b 43906	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 2737. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝑥 = 𝑥
Theorem	frege54cor1b 43907	Reflexive equality. (Contributed by RP, 24-Dec-2019.)
⊢ [𝑥 / 𝑦]𝑦 = 𝑥
Theorem	frege55lem1b 43908*	Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.)
⊢ ((𝜑 → [𝑥 / 𝑦]𝑦 = 𝑧) → (𝜑 → 𝑥 = 𝑧))
Theorem	frege55lem2b 43909	Lemma for frege55b 43910. 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)
Theorem	frege55b 43910	Lemma for frege57b 43912. Proposition 55 of [Frege1879] p. 50. Note that eqtr2 2761 incorporates eqcom 2744 which is stronger than this proposition which is identical to equcomi 2016. Is it possible that Frege tricked himself into assuming what he was out to prove? (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
Theorem	frege56b 43911	Lemma for frege57b 43912. 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)))
Theorem	frege57b 43912	Analogue of frege57aid 43885. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 → [𝑥 / 𝑧]𝜑))
Theorem	axfrege58b 43913	If ∀𝑥𝜑 is affirmed, [𝑦 / 𝑥]𝜑 cannot be denied. Identical to stdpc4 2068. Justification for ax-frege58b 43914. (Contributed by RP, 28-Mar-2020.)
⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
Axiom	ax-frege58b 43914	If ∀𝑥𝜑 is affirmed, [𝑦 / 𝑥]𝜑 cannot be denied. Identical to stdpc4 2068. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (New usage is discouraged.)
⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
Theorem	frege58bid 43915	If ∀𝑥𝜑 is affirmed, 𝜑 cannot be denied. Identical to sp 2183. See ax-frege58b 43914 and frege58c 43934 for versions which more closely track the original. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (Proof modification is discouraged.)
⊢ (∀𝑥𝜑 → 𝜑)
Theorem	frege58bcor 43916	Lemma for frege59b 43917. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
Theorem	frege59b 43917	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 43826 incorrectly referenced where frege30 43845 is in the original. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ([𝑦 / 𝑥]𝜑 → (¬ [𝑦 / 𝑥]𝜓 → ¬ ∀𝑥(𝜑 → 𝜓)))
Theorem	frege60b 43918	Swap antecedents of ax-frege58b 43914. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝑦 / 𝑥]𝜓 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))
Theorem	frege61b 43919	Lemma for frege65b 43923. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (([𝑥 / 𝑦]𝜑 → 𝜓) → (∀𝑦𝜑 → 𝜓))
Theorem	frege62b 43920	A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2663 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ([𝑦 / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜓) → [𝑦 / 𝑥]𝜓))
Theorem	frege63b 43921	Lemma for frege91 43967. Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ([𝑦 / 𝑥]𝜑 → (𝜓 → (∀𝑥(𝜑 → 𝜒) → [𝑦 / 𝑥]𝜒)))
Theorem	frege64b 43922	Lemma for frege65b 43923. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜓) → (∀𝑦(𝜓 → 𝜒) → ([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜒)))
Theorem	frege65b 43923	A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2663 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(𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))
Theorem	frege66b 43924	Swap antecedents of frege65b 43923. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜒 → 𝜑) → ([𝑦 / 𝑥]𝜒 → [𝑦 / 𝑥]𝜓)))
Theorem	frege67b 43925	Lemma for frege68b 43926. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝑦 / 𝑥]𝜑)))
Theorem	frege68b 43926	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.)
⊢ ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝑦 / 𝑥]𝜑))
21.36.6.7  _Begriffsschrift_ Chapter II with equivalence of classes Begriffsschrift Chapter II with equivalence of classes (where they are sets).
Theorem	frege53c 43927	Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ([𝐴 / 𝑥]𝜑 → (𝐴 = 𝐵 → [𝐵 / 𝑥]𝜑))
Theorem	frege54cor1c 43928*	Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Revised by RP, 25-Apr-2020.)
⊢ 𝐴 ∈ 𝐶    ⇒   ⊢ [𝐴 / 𝑥]𝑥 = 𝐴
Theorem	frege55lem1c 43929*	Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.)
⊢ ((𝜑 → [𝐴 / 𝑥]𝑥 = 𝐵) → (𝜑 → 𝐴 = 𝐵))
Theorem	frege55lem2c 43930*	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 43931	Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝐴 → 𝐴 = 𝑥)
Theorem	frege56c 43932*	Lemma for frege57c 43933. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝐵 ∈ 𝐶    ⇒   ⊢ ((𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑)) → (𝐵 = 𝐴 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑)))
Theorem	frege57c 43933*	Swap order of implication in ax-frege52c 43901. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝐶    ⇒   ⊢ (𝐴 = 𝐵 → ([𝐵 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑))
Theorem	frege58c 43934	Principle related to sp 2183. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)
Theorem	frege59c 43935	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 43826 incorrectly referenced where frege30 43845 is in the original. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 → (¬ [𝐴 / 𝑥]𝜓 → ¬ ∀𝑥(𝜑 → 𝜓)))
Theorem	frege60c 43936	Swap antecedents of frege58c 43934. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜓 → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒)))
Theorem	frege61c 43937	Lemma for frege65c 43941. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (([𝐴 / 𝑥]𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	frege62c 43938	A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2663 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 ∈ 𝐵    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜓) → [𝐴 / 𝑥]𝜓))
Theorem	frege63c 43939	Analogue of frege63b 43921. Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 → (𝜓 → (∀𝑥(𝜑 → 𝜒) → [𝐴 / 𝑥]𝜒)))
Theorem	frege64c 43940	Lemma for frege65c 43941. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (([𝐶 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝐶 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒)))
Theorem	frege65c 43941	A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2663 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 ∈ 𝐵    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒)))
Theorem	frege66c 43942	Swap antecedents of frege65c 43941. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜒 → 𝜑) → ([𝐴 / 𝑥]𝜒 → [𝐴 / 𝑥]𝜓)))
Theorem	frege67c 43943	Lemma for frege68c 43944. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝐴 / 𝑥]𝜑)))
Theorem	frege68c 43944	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 ∈ 𝐵    ⇒   ⊢ ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝐴 / 𝑥]𝜑))
21.36.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence (𝑅 “ 𝐴) ⊆ 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 43973 and frege109 43985. dffrege69 43945 through frege75 43951 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 43945*	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(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ 𝑅 hereditary 𝐴)
Theorem	frege70 43946*	Lemma for frege72 43948. Proposition 70 of [Frege1879] p. 58. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑉    ⇒   ⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → ∀𝑦(𝑋𝑅𝑦 → 𝑦 ∈ 𝐴)))
Theorem	frege71 43947*	Lemma for frege72 43948. Proposition 71 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑉    ⇒   ⊢ ((∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)) → (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴))))
Theorem	frege72 43948	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 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    ⇒   ⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)))
Theorem	frege73 43949	Lemma for frege87 43963. Proposition 73 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    ⇒   ⊢ ((𝑅 hereditary 𝐴 → 𝑋 ∈ 𝐴) → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)))
Theorem	frege74 43950	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 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    ⇒   ⊢ (𝑋 ∈ 𝐴 → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)))
Theorem	frege75 43951*	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(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) → 𝑅 hereditary 𝐴)
21.36.6.9  _Begriffsschrift_ Chapter III Following in a sequence 𝑝(t+‘𝑅)𝑐 means 𝑐 follows 𝑝 in the 𝑅-sequence. dffrege76 43952 through frege98 43974 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 43952*	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 ∈ 𝑈    &   ⊢ 𝐸 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    ⇒   ⊢ (∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝐸 ∈ 𝑓)) ↔ 𝐵(t+‘𝑅)𝐸)
Theorem	frege77 43953*	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 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (𝑋(t+‘𝑅)𝑌 → (𝑅 hereditary 𝐴 → (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴) → 𝑌 ∈ 𝐴)))
Theorem	frege78 43954*	Commuted form of frege77 43953. Proposition 78 of [Frege1879] p. 63. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (𝑅 hereditary 𝐴 → (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴) → (𝑋(t+‘𝑅)𝑌 → 𝑌 ∈ 𝐴)))
Theorem	frege79 43955*	Distributed form of frege78 43954. Proposition 79 of [Frege1879] p. 63. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ ((𝑅 hereditary 𝐴 → ∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴)) → (𝑅 hereditary 𝐴 → (𝑋(t+‘𝑅)𝑌 → 𝑌 ∈ 𝐴)))
Theorem	frege80 43956*	Add additional condition to both clauses of frege79 43955. Proposition 80 of [Frege1879] p. 63. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ ((𝑋 ∈ 𝐴 → (𝑅 hereditary 𝐴 → ∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴))) → (𝑋 ∈ 𝐴 → (𝑅 hereditary 𝐴 → (𝑋(t+‘𝑅)𝑌 → 𝑌 ∈ 𝐴))))
Theorem	frege81 43957	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 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (𝑋 ∈ 𝐴 → (𝑅 hereditary 𝐴 → (𝑋(t+‘𝑅)𝑌 → 𝑌 ∈ 𝐴)))
Theorem	frege82 43958	Closed-form deduction based on frege81 43957. Proposition 82 of [Frege1879] p. 64. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ ((𝜑 → 𝑋 ∈ 𝐴) → (𝑅 hereditary 𝐴 → (𝜑 → (𝑋(t+‘𝑅)𝑌 → 𝑌 ∈ 𝐴))))
Theorem	frege83 43959	Apply commuted form of frege81 43957 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 ∈ 𝑆    &   ⊢ 𝑌 ∈ 𝑇    &   ⊢ 𝑅 ∈ 𝑈    &   ⊢ 𝐵 ∈ 𝑉    &   ⊢ 𝐶 ∈ 𝑊    ⇒   ⊢ (𝑅 hereditary (𝐵 ∪ 𝐶) → (𝑋 ∈ 𝐵 → (𝑋(t+‘𝑅)𝑌 → 𝑌 ∈ (𝐵 ∪ 𝐶))))
Theorem	frege84 43960	Commuted form of frege81 43957. Proposition 84 of [Frege1879] p. 65. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋(t+‘𝑅)𝑌 → 𝑌 ∈ 𝐴)))
Theorem	frege85 43961*	Commuted form of frege77 43953. Proposition 85 of [Frege1879] p. 66. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (𝑋(t+‘𝑅)𝑌 → (∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → (𝑅 hereditary 𝐴 → 𝑌 ∈ 𝐴)))
Theorem	frege86 43962*	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 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (((𝑅 hereditary 𝐴 → 𝑌 ∈ 𝐴) → (𝑅 hereditary 𝐴 → (𝑌𝑅𝑍 → 𝑍 ∈ 𝐴))) → (𝑋(t+‘𝑅)𝑌 → (∀𝑤(𝑋𝑅𝑤 → 𝑤 ∈ 𝐴) → (𝑅 hereditary 𝐴 → (𝑌𝑅𝑍 → 𝑍 ∈ 𝐴)))))
Theorem	frege87 43963*	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 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑍 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝑆    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (𝑋(t+‘𝑅)𝑌 → (∀𝑤(𝑋𝑅𝑤 → 𝑤 ∈ 𝐴) → (𝑅 hereditary 𝐴 → (𝑌𝑅𝑍 → 𝑍 ∈ 𝐴))))
Theorem	frege88 43964*	Commuted form of frege87 43963. Proposition 88 of [Frege1879] p. 67. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑍 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝑆    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (𝑌𝑅𝑍 → (𝑋(t+‘𝑅)𝑌 → (∀𝑤(𝑋𝑅𝑤 → 𝑤 ∈ 𝐴) → (𝑅 hereditary 𝐴 → 𝑍 ∈ 𝐴))))
Theorem	frege89 43965*	One direction of dffrege76 43952. Proposition 89 of [Frege1879] p. 68. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    ⇒   ⊢ (∀𝑓(𝑅 hereditary 𝑓 → (∀𝑤(𝑋𝑅𝑤 → 𝑤 ∈ 𝑓) → 𝑌 ∈ 𝑓)) → 𝑋(t+‘𝑅)𝑌)
Theorem	frege90 43966*	Add antecedent to frege89 43965. Proposition 90 of [Frege1879] p. 68. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    ⇒   ⊢ ((𝜑 → ∀𝑓(𝑅 hereditary 𝑓 → (∀𝑤(𝑋𝑅𝑤 → 𝑤 ∈ 𝑓) → 𝑌 ∈ 𝑓))) → (𝜑 → 𝑋(t+‘𝑅)𝑌))
Theorem	frege91 43967	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 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    ⇒   ⊢ (𝑋𝑅𝑌 → 𝑋(t+‘𝑅)𝑌)
Theorem	frege92 43968	Inference from frege91 43967. Proposition 92 of [Frege1879] p. 69. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    ⇒   ⊢ (𝑋 = 𝑍 → (𝑋𝑅𝑌 → 𝑍(t+‘𝑅)𝑌))
Theorem	frege93 43969*	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 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    ⇒   ⊢ (∀𝑓(∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝑓) → (𝑅 hereditary 𝑓 → 𝑌 ∈ 𝑓)) → 𝑋(t+‘𝑅)𝑌)
Theorem	frege94 43970*	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 ∈ 𝑈    &   ⊢ 𝑍 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    ⇒   ⊢ ((𝑌𝑅𝑍 → (𝑋(t+‘𝑅)𝑌 → ∀𝑓(∀𝑤(𝑋𝑅𝑤 → 𝑤 ∈ 𝑓) → (𝑅 hereditary 𝑓 → 𝑍 ∈ 𝑓)))) → (𝑌𝑅𝑍 → (𝑋(t+‘𝑅)𝑌 → 𝑋(t+‘𝑅)𝑍)))
Theorem	frege95 43971	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 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑍 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝐴    ⇒   ⊢ (𝑌𝑅𝑍 → (𝑋(t+‘𝑅)𝑌 → 𝑋(t+‘𝑅)𝑍))
Theorem	frege96 43972	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 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑍 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝐴    ⇒   ⊢ (𝑋(t+‘𝑅)𝑌 → (𝑌𝑅𝑍 → 𝑋(t+‘𝑅)𝑍))
Theorem	frege97 43973	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 ∈ 𝑈    &   ⊢ 𝑅 ∈ 𝑊    ⇒   ⊢ 𝑅 hereditary ((t+‘𝑅) “ {𝑋})
Theorem	frege98 43974	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 ∈ 𝐴    &   ⊢ 𝑌 ∈ 𝐵    &   ⊢ 𝑍 ∈ 𝐶    &   ⊢ 𝑅 ∈ 𝐷    ⇒   ⊢ (𝑋(t+‘𝑅)𝑌 → (𝑌(t+‘𝑅)𝑍 → 𝑋(t+‘𝑅)𝑍))
21.36.6.10  _Begriffsschrift_ Chapter III Member of sequence 𝑝((t+‘𝑅) ∪ I )𝑐 means 𝑐 is a member of the 𝑅 -sequence beginning with 𝑝 and 𝑝 is a member of the 𝑅 -sequence ending with 𝑐. dffrege99 43975 through frege114 43990 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 43975	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 ∈ 𝑈    ⇒   ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍)
Theorem	frege100 43976	One direction of dffrege99 43975. Proposition 100 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑍 ∈ 𝑈    ⇒   ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋))
Theorem	frege101 43977	Lemma for frege102 43978. Proposition 101 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑍 ∈ 𝑈    ⇒   ⊢ ((𝑍 = 𝑋 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)) → ((𝑋(t+‘𝑅)𝑍 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉))))
Theorem	frege102 43978	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 ∈ 𝐴    &   ⊢ 𝑍 ∈ 𝐵    &   ⊢ 𝑉 ∈ 𝐶    &   ⊢ 𝑅 ∈ 𝐷    ⇒   ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉))
Theorem	frege103 43979	Proposition 103 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑍 ∈ 𝑉    ⇒   ⊢ ((𝑍 = 𝑋 → 𝑋 = 𝑍) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 = 𝑍)))
Theorem	frege104 43980	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 ∈ 𝑉    ⇒   ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 = 𝑍))
Theorem	frege105 43981	Proposition 105 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑍 ∈ 𝑉    ⇒   ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) → 𝑋((t+‘𝑅) ∪ I )𝑍)
Theorem	frege106 43982	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 ∈ 𝑉    ⇒   ⊢ (𝑋(t+‘𝑅)𝑍 → 𝑋((t+‘𝑅) ∪ I )𝑍)
Theorem	frege107 43983	Proposition 107 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑉 ∈ 𝐴    ⇒   ⊢ ((𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → 𝑍(t+‘𝑅)𝑉)) → (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → 𝑍((t+‘𝑅) ∪ I )𝑉)))
Theorem	frege108 43984	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 ∈ 𝐴    &   ⊢ 𝑌 ∈ 𝐵    &   ⊢ 𝑉 ∈ 𝐶    &   ⊢ 𝑅 ∈ 𝐷    ⇒   ⊢ (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → 𝑍((t+‘𝑅) ∪ I )𝑉))
Theorem	frege109 43985	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 ∈ 𝑈    &   ⊢ 𝑅 ∈ 𝑉    ⇒   ⊢ 𝑅 hereditary (((t+‘𝑅) ∪ I ) “ {𝑋})
Theorem	frege110 43986*	Proposition 110 of [Frege1879] p. 75. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝐴    &   ⊢ 𝑌 ∈ 𝐵    &   ⊢ 𝑀 ∈ 𝐶    &   ⊢ 𝑅 ∈ 𝐷    ⇒   ⊢ (∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀))
Theorem	frege111 43987	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 ∈ 𝐴    &   ⊢ 𝑌 ∈ 𝐵    &   ⊢ 𝑉 ∈ 𝐶    &   ⊢ 𝑅 ∈ 𝐷    ⇒   ⊢ (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → (¬ 𝑉(t+‘𝑅)𝑍 → 𝑍((t+‘𝑅) ∪ I )𝑉)))
Theorem	frege112 43988	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 ∈ 𝑉    ⇒   ⊢ (𝑍 = 𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍)
Theorem	frege113 43989	Proposition 113 of [Frege1879] p. 76. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑍 ∈ 𝑉    ⇒   ⊢ ((𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑍 = 𝑋)) → (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍)))
Theorem	frege114 43990	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 ∈ 𝑈    &   ⊢ 𝑍 ∈ 𝑉    ⇒   ⊢ (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍))
21.36.6.11  _Begriffsschrift_ Chapter III Single-valued procedures Fun ◡◡𝑅 means the relationship content of procedure 𝑅 is single-valued. The double converse allows to simply apply this syntax in place of Frege's even though the original never explicitly limited discussion of propositional statements which vary on two variables to relations. dffrege115 43991 through frege133 44009 develop this and how functions relate to transitive and transitive-reflexive closures.
Theorem	dffrege115 43991*	If from 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.)
⊢ (∀𝑐∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ Fun ◡◡𝑅)
Theorem	frege116 43992*	One direction of dffrege115 43991. Proposition 116 of [Frege1879] p. 77. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    ⇒   ⊢ (Fun ◡◡𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
Theorem	frege117 43993*	Lemma for frege118 43994. Proposition 117 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    ⇒   ⊢ ((∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋))) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋))))
Theorem	frege118 43994*	Simplified application of one direction of dffrege115 43991. Proposition 118 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    ⇒   ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋)))
Theorem	frege119 43995*	Lemma for frege120 43996. Proposition 119 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    ⇒   ⊢ ((∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋) → (𝑌𝑅𝐴 → 𝐴 = 𝑋)) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋))))
Theorem	frege120 43996	Simplified application of one direction of dffrege115 43991. Proposition 120 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝐴 ∈ 𝑊    ⇒   ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋)))
Theorem	frege121 43997	Lemma for frege122 43998. Proposition 121 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝐴 ∈ 𝑊    ⇒   ⊢ ((𝐴 = 𝑋 → 𝑋((t+‘𝑅) ∪ I )𝐴) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝑋((t+‘𝑅) ∪ I )𝐴))))
Theorem	frege122 43998	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 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝐴 ∈ 𝑊    ⇒   ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝑋((t+‘𝑅) ∪ I )𝐴)))
Theorem	frege123 43999*	Lemma for frege124 44000. Proposition 123 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    ⇒   ⊢ ((∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀)) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀))))
Theorem	frege124 44000	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 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑀 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝑆    ⇒   ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀)))