HomeTheorem List (p. 451 of 512)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | frege63b 45001 | Lemma for frege91 45047. 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 45002 | Lemma for frege65b 45003. 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 45003 |
A kind of Aristotelian inference. This judgement replaces the mode of
inference barbara 2687 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 45004 | Swap antecedents of frege65b 45003. 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 45005 | Lemma for frege68b 45006. 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 45006 | 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)) | ||
Begriffsschrift Chapter II with equivalence of classes (where they are sets). | ||
| Theorem | frege53c 45007 | 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 45008* | Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Revised by RP, 25-Apr-2020.) |
| ⊢ 𝐴 ∈ 𝐶 ⇒ ⊢ [𝐴 / 𝑥]𝑥 = 𝐴 | ||
| Theorem | frege55lem1c 45009* | Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.) |
| ⊢ ((𝜑 → [𝐴 / 𝑥]𝑥 = 𝐵) → (𝜑 → 𝐴 = 𝐵)) | ||
| Theorem | frege55lem2c 45010* | 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 45011 | Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
| ⊢ (𝑥 = 𝐴 → 𝐴 = 𝑥) | ||
| Theorem | frege56c 45012* | Lemma for frege57c 45013. 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 45013* | Swap order of implication in ax-frege52c 44981. 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 45014 | Principle related to sp 2219. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
| ⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑) | ||
| Theorem | frege59c 45015 |
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 44906 incorrectly referenced where frege30 44925 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 45016 | Swap antecedents of frege58c 45014. 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 45017 | Lemma for frege65c 45021. 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 45018 | A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2687 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 45019 | Analogue of frege63b 45001. 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 45020 | Lemma for frege65c 45021. 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 45021 | A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2687 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 45022 | Swap antecedents of frege65c 45021. 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 45023 | Lemma for frege68c 45024. 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 45024 | 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)) | ||
(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 45053 and frege109 45065. dffrege69 45025 through frege75 45031 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 45025* | 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 45026* | Lemma for frege72 45028. 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 45027* | Lemma for frege72 45028. 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 45028 | 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 45029 | Lemma for frege87 45043. 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 45030 | 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 45031* | 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) | ||
𝑝(t+‘𝑅)𝑐 means 𝑐 follows 𝑝 in the 𝑅-sequence. dffrege76 45032 through frege98 45054 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 45032* |
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 45033* | 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 45034* | Commuted form of frege77 45033. 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 45035* | Distributed form of frege78 45034. 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 45036* | Add additional condition to both clauses of frege79 45035. 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 45037 | 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 45038 | Closed-form deduction based on frege81 45037. 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 45039 | Apply commuted form of frege81 45037 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 45040 | Commuted form of frege81 45037. 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 45041* | Commuted form of frege77 45033. 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 45042* | 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 45043* | 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 45044* | Commuted form of frege87 45043. 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 45045* | One direction of dffrege76 45032. 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 45046* | Add antecedent to frege89 45045. 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 45047 | 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 45048 | Inference from frege91 45047. 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 45049* | 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 45050* | 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 45051 | 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 45052 | 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 45053 |
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 45054 | 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)) | ||
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 45055 through frege114 45070 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 45055 | 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 45056 | One direction of dffrege99 45055. 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 45057 | Lemma for frege102 45058. 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 45058 | 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 45059 | 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 45060 |
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 45061 | 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 45062 | 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 45063 | 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 45064 | 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 45065 | 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 45066* | 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 45067 | 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 45068 | 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 45069 | 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 45070 | 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)) | ||
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 45071 through frege133 45089 develop this and how functions relate to transitive and transitive-reflexive closures. | ||
| Theorem | dffrege115 45071* | 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 45072* | One direction of dffrege115 45071. 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 45073* | Lemma for frege118 45074. 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 45074* | Simplified application of one direction of dffrege115 45071. 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 45075* | Lemma for frege120 45076. 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 45076 | Simplified application of one direction of dffrege115 45071. 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 45077 | Lemma for frege122 45078. 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 45078 | 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 45079* | Lemma for frege124 45080. 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 45080 | 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))) | ||
| Theorem | frege125 45081 | Lemma for frege126 45082. Proposition 125 of [Frege1879] p. 81. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.) |
| ⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑀 ∈ 𝑊 & ⊢ 𝑅 ∈ 𝑆 ⇒ ⊢ ((𝑋((t+‘𝑅) ∪ I )𝑀 → (¬ 𝑋(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑋)) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀 → (¬ 𝑋(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑋))))) | ||
| Theorem | frege126 45082 | 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 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑀 ∈ 𝑊 & ⊢ 𝑅 ∈ 𝑆 ⇒ ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀 → (¬ 𝑋(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑋)))) | ||
| Theorem | frege127 45083 | Communte antecedents of frege126 45082. Proposition 127 of [Frege1879] p. 82. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.) |
| ⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑀 ∈ 𝑊 & ⊢ 𝑅 ∈ 𝑆 ⇒ ⊢ (Fun ◡◡𝑅 → (𝑌(t+‘𝑅)𝑀 → (𝑌𝑅𝑋 → (¬ 𝑋(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑋)))) | ||
| Theorem | frege128 45084 | Lemma for frege129 45085. Proposition 128 of [Frege1879] p. 83. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.) |
| ⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑀 ∈ 𝑊 & ⊢ 𝑅 ∈ 𝑆 ⇒ ⊢ ((𝑀((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑋 → (¬ 𝑋(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑋))) → (Fun ◡◡𝑅 → ((¬ 𝑌(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑌) → (𝑌𝑅𝑋 → (¬ 𝑋(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑋))))) | ||
| Theorem | frege129 45085 | If the procedure 𝑅 is single-valued and 𝑌 belongs to the 𝑅 -sequence beginning 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 beginning 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 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑀 ∈ 𝑊 & ⊢ 𝑅 ∈ 𝑆 ⇒ ⊢ (Fun ◡◡𝑅 → ((¬ 𝑌(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑌) → (𝑌𝑅𝑋 → (¬ 𝑋(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑋)))) | ||
| Theorem | frege130 45086* | Lemma for frege131 45087. Proposition 130 of [Frege1879] p. 84. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.) |
| ⊢ 𝑀 ∈ 𝑈 & ⊢ 𝑅 ∈ 𝑉 ⇒ ⊢ ((∀𝑏((¬ 𝑏(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑏) → ∀𝑎(𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑎))) → 𝑅 hereditary ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))) → (Fun ◡◡𝑅 → 𝑅 hereditary ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})))) | ||
| Theorem | frege131 45087 | If the procedure 𝑅 is single-valued, then the property of belonging to the 𝑅-sequence beginning 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 ∈ 𝑈 & ⊢ 𝑅 ∈ 𝑉 ⇒ ⊢ (Fun ◡◡𝑅 → 𝑅 hereditary ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))) | ||
| Theorem | frege132 45088 | Lemma for frege133 45089. Proposition 132 of [Frege1879] p. 86. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.) |
| ⊢ 𝑀 ∈ 𝑈 & ⊢ 𝑅 ∈ 𝑉 ⇒ ⊢ ((𝑅 hereditary ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) → (𝑋(t+‘𝑅)𝑀 → (𝑋(t+‘𝑅)𝑌 → (¬ 𝑌(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑌)))) → (Fun ◡◡𝑅 → (𝑋(t+‘𝑅)𝑀 → (𝑋(t+‘𝑅)𝑌 → (¬ 𝑌(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑌))))) | ||
| Theorem | frege133 45089 | 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 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑀 ∈ 𝑊 & ⊢ 𝑅 ∈ 𝑆 ⇒ ⊢ (Fun ◡◡𝑅 → (𝑋(t+‘𝑅)𝑀 → (𝑋(t+‘𝑅)𝑌 → (¬ 𝑌(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑌)))) | ||
See Seifert and Threlfall: A Textbook Of Topology (1980) which is an English translation of Lehrbuch der Topologie (1934). | ||
Because ((2o ↑m 𝐵) ↑m 𝐴) ≈ (2o ↑m (𝐴 × 𝐵)) ≈ ((2o ↑m 𝐴) ↑m 𝐵) is an instance of the law of exponents: ((𝐶 ↑m 𝐵) ↑m 𝐴) ≈ (𝐶 ↑m (𝐴 × 𝐵)) ≈ ((𝐶 ↑m 𝐴) ↑m 𝐵) we are led to see that (𝒫 𝐵 ↑m 𝐴) ≈ 𝒫 (𝐴 × 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 45090 | 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 45099 for a demonstration of a natural one-to-one onto mapping. (Contributed by RP, 27-Apr-2021.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵 ↑m 𝐴)) | ||
| Theorem | enrelmapr 45091 | 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 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐴 ↑m 𝐵)) | ||
| Theorem | enmappw 45092 | 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 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐵 ↑m 𝐴) ≈ (𝒫 𝐴 ↑m 𝐵)) | ||
| Theorem | enmappwid 45093 | 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 ∈ 𝑉 → (𝒫 𝐴 ↑m 𝒫 𝐴) ≈ (𝒫 𝒫 𝐴 ↑m 𝐴)) | ||
| Theorem | rfovd 45094* | Value of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵. (Contributed by RP, 25-Apr-2021.) |
| ⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦}))) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐴𝑂𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))) | ||
| Theorem | rfovfvd 45095* | Value of the operator, (𝐴𝑂𝐵), 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 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦}))) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ 𝒫 (𝐴 × 𝐵)) & ⊢ 𝐹 = (𝐴𝑂𝐵) ⇒ ⊢ (𝜑 → (𝐹‘𝑅) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦})) | ||
| Theorem | rfovfvfvd 45096* | Value of the operator, (𝐴𝑂𝐵), 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 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦}))) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ 𝒫 (𝐴 × 𝐵)) & ⊢ 𝐹 = (𝐴𝑂𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ 𝐺 = (𝐹‘𝑅) ⇒ ⊢ (𝜑 → (𝐺‘𝑋) = {𝑦 ∈ 𝐵 ∣ 𝑋𝑅𝑦}) | ||
| Theorem | rfovcnvf1od 45097* | Properties of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵. (Contributed by RP, 27-Apr-2021.) |
| ⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦}))) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ 𝐹 = (𝐴𝑂𝐵) ⇒ ⊢ (𝜑 → (𝐹:𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝐴) ∧ ◡𝐹 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}))) | ||
| Theorem | rfovcnvd 45098* | Value of the converse of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵. (Contributed by RP, 27-Apr-2021.) |
| ⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦}))) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ 𝐹 = (𝐴𝑂𝐵) ⇒ ⊢ (𝜑 → ◡𝐹 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})) | ||
| Theorem | rfovf1od 45099* | The value of the operator, (𝐴𝑂𝐵), 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 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦}))) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ 𝐹 = (𝐴𝑂𝐵) ⇒ ⊢ (𝜑 → 𝐹:𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝐴)) | ||
| Theorem | rfovcnvfvd 45100* | Value of the converse of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵, evaluated at function 𝐺. (Contributed by RP, 27-Apr-2021.) |
| ⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦}))) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ 𝐹 = (𝐴𝑂𝐵) & ⊢ (𝜑 → 𝐺 ∈ (𝒫 𝐵 ↑m 𝐴)) ⇒ ⊢ (𝜑 → (◡𝐹‘𝐺) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐺‘𝑥))}) | ||
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