Home | Metamath
Proof Explorer Theorem List (p. 459 of 461) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | Metamath Proof Explorer
(1-28865) |
Hilbert Space Explorer
(28866-30388) |
Users' Mathboxes
(30389-46009) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | iscnrm3lem6 45801* | Lemma for iscnrm3lem7 45802. (Contributed by Zhi Wang, 5-Sep-2024.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊) ∧ 𝜓) → 𝜒) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑊 𝜓 → 𝜒)) | ||
Theorem | iscnrm3lem7 45802* | Lemma for iscnrm3rlem8 45810 and iscnrm3llem2 45813 involving restricted existential quantifications. (Contributed by Zhi Wang, 5-Sep-2024.) |
⊢ (𝑧 = 𝑍 → (𝜒 ↔ 𝜃)) & ⊢ (𝑤 = 𝑊 → (𝜃 ↔ 𝜏)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → (𝑍 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ∧ 𝜏)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜒)) | ||
Theorem | iscnrm3rlem1 45803 | Lemma for iscnrm3rlem2 45804. The hypothesis could be generalized to (𝜑 → (𝑆 ∖ 𝑇) ⊆ 𝑋). (Contributed by Zhi Wang, 5-Sep-2024.) |
⊢ (𝜑 → 𝑆 ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝑆 ∖ 𝑇) = (𝑆 ∩ (𝑋 ∖ (𝑆 ∩ 𝑇)))) | ||
Theorem | iscnrm3rlem2 45804 | Lemma for iscnrm3rlem3 45805. (Contributed by Zhi Wang, 5-Sep-2024.) |
⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽) ⇒ ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∖ 𝑇) ∈ (Clsd‘(𝐽 ↾_{t} (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ 𝑇))))) | ||
Theorem | iscnrm3rlem3 45805 | Lemma for iscnrm3r 45811. The designed subspace is a subset of the original set; the two sets are closed sets in the subspace. (Contributed by Zhi Wang, 5-Sep-2024.) |
⊢ ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽)) → ((∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝒫 ∪ 𝐽 ∧ (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘(𝐽 ↾_{t} (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ∈ (Clsd‘(𝐽 ↾_{t} (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))))) | ||
Theorem | iscnrm3rlem4 45806 | Lemma for iscnrm3rlem8 45810. Given two disjoint subsets 𝑆 and 𝑇 of the underlying set of a topology 𝐽, if 𝑁 is a superset of (((cls‘𝐽)‘𝑆) ∖ 𝑇), then it is a superset of 𝑆. (Contributed by Zhi Wang, 5-Sep-2024.) |
⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽) & ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅) & ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∖ 𝑇) ⊆ 𝑁) ⇒ ⊢ (𝜑 → 𝑆 ⊆ 𝑁) | ||
Theorem | iscnrm3rlem5 45807 | Lemma for iscnrm3rlem6 45808. (Contributed by Zhi Wang, 5-Sep-2024.) |
⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽) & ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽) ⇒ ⊢ (𝜑 → (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝐽) | ||
Theorem | iscnrm3rlem6 45808 | Lemma for iscnrm3rlem7 45809. (Contributed by Zhi Wang, 5-Sep-2024.) |
⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽) & ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽) & ⊢ (𝜑 → 𝑂 ⊆ (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ⇒ ⊢ (𝜑 → (𝑂 ∈ (𝐽 ↾_{t} (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ↔ 𝑂 ∈ 𝐽)) | ||
Theorem | iscnrm3rlem7 45809 | Lemma for iscnrm3rlem8 45810. Open neighborhoods in the subspace topology are open neighborhoods in the original topology given that the subspace is an open set in the original topology. (Contributed by Zhi Wang, 5-Sep-2024.) |
⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽) & ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽) & ⊢ (𝜑 → 𝑂 ∈ (𝐽 ↾_{t} (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ⇒ ⊢ (𝜑 → 𝑂 ∈ 𝐽) | ||
Theorem | iscnrm3rlem8 45810* | Lemma for iscnrm3r 45811. Disjoint open neighborhoods in the subspace topology are disjoint open neighborhoods in the original topology given that the subspace is an open set in the original topology. Therefore, given any two sets separated in the original topology and separated by open neighborhoods in the subspace topology, they must be separated by open neighborhoods in the original topology. (Contributed by Zhi Wang, 5-Sep-2024.) |
⊢ ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) → (∃𝑙 ∈ (𝐽 ↾_{t} (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽 ↾_{t} (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))) | ||
Theorem | iscnrm3r 45811* | Lemma for iscnrm3 45815. If all subspaces of a topology are normal, i.e., two disjoint closed sets can be separated by open neighborhoods, then in the original topology two separated sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.) |
⊢ (𝐽 ∈ Top → (∀𝑧 ∈ 𝒫 ∪ 𝐽∀𝑐 ∈ (Clsd‘(𝐽 ↾_{t} 𝑧))∀𝑑 ∈ (Clsd‘(𝐽 ↾_{t} 𝑧))((𝑐 ∩ 𝑑) = ∅ → ∃𝑙 ∈ (𝐽 ↾_{t} 𝑧)∃𝑘 ∈ (𝐽 ↾_{t} 𝑧)(𝑐 ⊆ 𝑙 ∧ 𝑑 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → ((𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) → (((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))) | ||
Theorem | iscnrm3llem1 45812 | Lemma for iscnrm3l 45814. Closed sets in the subspace are subsets of the underlying set of the original topology. (Contributed by Zhi Wang, 4-Sep-2024.) |
⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾_{t} 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾_{t} 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → (𝐶 ∈ 𝒫 ∪ 𝐽 ∧ 𝐷 ∈ 𝒫 ∪ 𝐽)) | ||
Theorem | iscnrm3llem2 45813* | Lemma for iscnrm3l 45814. If there exist disjoint open neighborhoods in the orignal topology for two disjoint closed sets in a subspace, then they can be separated by open neighborhoods in the subspace topology. (Could shorten proof with ssin0 42181.) (Contributed by Zhi Wang, 5-Sep-2024.) |
⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾_{t} 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾_{t} 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → (∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅) → ∃𝑙 ∈ (𝐽 ↾_{t} 𝑍)∃𝑘 ∈ (𝐽 ↾_{t} 𝑍)(𝐶 ⊆ 𝑙 ∧ 𝐷 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅))) | ||
Theorem | iscnrm3l 45814* | Lemma for iscnrm3 45815. Given a topology 𝐽, if two separated sets can be separated by open neighborhoods, then all subspaces of the topology 𝐽 are normal, i.e., two disjoint closed sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.) |
⊢ (𝐽 ∈ Top → (∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾_{t} 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾_{t} 𝑍))) → ((𝐶 ∩ 𝐷) = ∅ → ∃𝑙 ∈ (𝐽 ↾_{t} 𝑍)∃𝑘 ∈ (𝐽 ↾_{t} 𝑍)(𝐶 ⊆ 𝑙 ∧ 𝐷 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅))))) | ||
Theorem | iscnrm3 45815* | A completely normal topology is a topology in which two separated sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.) |
⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))) | ||
Theorem | iscnrm3v 45816* | A topology is completely normal iff two separated sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 10-Sep-2024.) |
⊢ (𝐽 ∈ Top → (𝐽 ∈ CNrm ↔ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))) | ||
Theorem | iscnrm4 45817* | A completely normal topology is a topology in which two separated sets can be separated by neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.) |
⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ ((nei‘𝐽)‘𝑠)∃𝑚 ∈ ((nei‘𝐽)‘𝑡)(𝑛 ∩ 𝑚) = ∅))) | ||
Theorem | isprsd 45818* | Property of being a preordered set (deduction form). (Contributed by Zhi Wang, 18-Sep-2024.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → ≤ = (le‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐾 ∈ Proset ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)))) | ||
Theorem | catprslem 45819* | Lemma for catprs 45820. (Contributed by Zhi Wang, 18-Sep-2024.) |
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅)) | ||
Theorem | catprs 45820* | A preorder can be extracted from a category. See catprs2 45821 for more details. (Contributed by Zhi Wang, 18-Sep-2024.) |
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ Cat) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍))) | ||
Theorem | catprs2 45821* | A category equipped with the induced preorder, where an object 𝑥 is defined to be "less than or equal to" 𝑦 iff there is a morphism from 𝑥 to 𝑦, is a preordered set, or a proset. The category might not be thin. See catprsc 45822 and catprsc2 45823 for constructions satisfying the hypothesis "catprs.1". See catprs 45820 for a more primitive version. See prsthinc 45846 for constructing a thin category from a proset. (Contributed by Zhi Wang, 18-Sep-2024.) |
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → ≤ = (le‘𝐶)) ⇒ ⊢ (𝜑 → 𝐶 ∈ Proset ) | ||
Theorem | catprsc 45822* | A construction of the preorder induced by a category. See catprs2 45821 for details. See also catprsc2 45823 for an alternate construction. (Contributed by Zhi Wang, 18-Sep-2024.) |
⊢ (𝜑 → ≤ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)}) ⇒ ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅)) | ||
Theorem | catprsc2 45823* | An alternate construction of the preorder induced by a category. See catprs2 45821 for details. See also catprsc 45822 for a different construction. The two constructions are different because df-cat 17054 does not require the domain of 𝐻 to be 𝐵 × 𝐵. (Contributed by Zhi Wang, 23-Sep-2024.) |
⊢ (𝜑 → ≤ = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐻𝑦) ≠ ∅}) ⇒ ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅)) | ||
Theorem | idmon 45824 | An identity arrow, or an identity morphism, is a monomorphism. (Contributed by Zhi Wang, 21-Sep-2024.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 𝑀 = (Mono‘𝐶) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝑀𝑋)) | ||
Theorem | idepi 45825 | An identity arrow, or an identity morphism, is an epimorphism. (Contributed by Zhi Wang, 21-Sep-2024.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 𝐸 = (Epi‘𝐶) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐸𝑋)) | ||
Syntax | cthinc 45826 | Extend class notation with the class of thin categories. |
class ThinCat | ||
Definition | df-thinc 45827* | Definition of the class of thin categories, or posetal categories, whose hom-sets each contain at most one morphism. Example 3.26(2) of [Adamek] p. 33. "ThinCat" was taken instead of "PosCat" because the latter might mean the category of posets. (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ ThinCat = {𝑐 ∈ Cat ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)} | ||
Theorem | isthinc 45828* | The predicate "is a thin category". (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))) | ||
Theorem | isthinc2 45829* | A thin category is a category in which all hom-sets have cardinality less than or equal to the cardinality of 1_{o}. (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ≼ 1_{o})) | ||
Theorem | isthinc3 45830* | A thin category is a category in which, given a pair of objects 𝑥 and 𝑦 and any two morphisms 𝑓, 𝑔 from 𝑥 to 𝑦, the morphisms are equal. (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑥𝐻𝑦)𝑓 = 𝑔)) | ||
Theorem | isthincd2lem1 45831* | Lemma for isthincd2 45839 and thincmo2 45832. (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)) ⇒ ⊢ (𝜑 → 𝐹 = 𝐺) | ||
Theorem | thincmo2 45832 | Morphisms in the same hom-set are identical. (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌)) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ ThinCat) ⇒ ⊢ (𝜑 → 𝐹 = 𝐺) | ||
Theorem | thincmo 45833* | There is at most one morphism in each hom-set. (Contributed by Zhi Wang, 21-Sep-2024.) |
⊢ (𝜑 → 𝐶 ∈ ThinCat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)) | ||
Theorem | thincmoALT 45834* | Alternate proof for thincmo 45833. (Contributed by Zhi Wang, 21-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 ∈ ThinCat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)) | ||
Theorem | thincmod 45835* | At most one morphism in each hom-set (deduction form). (Contributed by Zhi Wang, 21-Sep-2024.) |
⊢ (𝜑 → 𝐶 ∈ ThinCat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) ⇒ ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)) | ||
Theorem | thincn0eu 45836* | In a thin category, a hom-set being non-empty is equivalent to having a unique element. (Contributed by Zhi Wang, 21-Sep-2024.) |
⊢ (𝜑 → 𝐶 ∈ ThinCat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) ⇒ ⊢ (𝜑 → ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌))) | ||
Theorem | isthincd2lem2 45837* | Lemma for isthincd2 45839. (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) ⇒ ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ (𝑋𝐻𝑍)) | ||
Theorem | isthincd 45838* | The predicate "is a thin category" (deduction form). (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)) & ⊢ (𝜑 → 𝐶 ∈ Cat) ⇒ ⊢ (𝜑 → 𝐶 ∈ ThinCat) | ||
Theorem | isthincd2 45839* | The predicate "𝐶 is a thin category" without knowing 𝐶 is a category (deduction form). The identity arrow operator is also provided as a byproduct. (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)) & ⊢ (𝜑 → · = (comp‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜓 ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧)))) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 1 ∈ (𝑦𝐻𝑦)) & ⊢ ((𝜑 ∧ 𝜓) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) ⇒ ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ 1 ))) | ||
Theorem | thincc 45840 | A thin category is a category. (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ (𝐶 ∈ ThinCat → 𝐶 ∈ Cat) | ||
Theorem | thincssc 45841 | A thin category is a category. (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ ThinCat ⊆ Cat | ||
Theorem | 0thincg 45842 | Any structure with an empty set of objects is a thin category. (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ ThinCat) | ||
Theorem | 0thinc 45843 | The empty category (see 0cat 17075) is thin. (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ ∅ ∈ ThinCat | ||
Theorem | indthinc 45844* | An indiscrete category in which all hom-sets have exactly one morphism is a thin category. Constructed here is an indiscrete category where all morphisms are ∅. This is a special case of prsthinc 45846, where ≤ = (𝐵 × 𝐵). This theorem also implies a functor from the category of sets to the category of small categories. (Contributed by Zhi Wang, 17-Sep-2024.) (Proof shortened by Zhi Wang, 19-Sep-2024.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → ((𝐵 × 𝐵) × {1_{o}}) = (Hom ‘𝐶)) & ⊢ (𝜑 → ∅ = (comp‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅))) | ||
Theorem | indthincALT 45845* | An alternate proof for indthinc 45844 assuming more axioms including ax-pow 5242 and ax-un 7491. (Contributed by Zhi Wang, 17-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → ((𝐵 × 𝐵) × {1_{o}}) = (Hom ‘𝐶)) & ⊢ (𝜑 → ∅ = (comp‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅))) | ||
Theorem | prsthinc 45846* | Preordered sets as categories. Similar to example 3.3(4.d) of [Adamek] p. 24, but the hom-sets are not pairwise disjoint. One can define a functor from the category of prosets to the category of small thin categories. See catprs 45820 and catprs2 45821 for inducing a preorder from a category. Example 3.26(2) of [Adamek] p. 33 indicates that it induces a bijection from the equivalence class of isomorphic small thin categories to the equivalence class of order-isomorphic preordered sets. (Contributed by Zhi Wang, 18-Sep-2024.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → ( ≤ × {1_{o}}) = (Hom ‘𝐶)) & ⊢ (𝜑 → ∅ = (comp‘𝐶)) & ⊢ (𝜑 → ≤ = (le‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ Proset ) ⇒ ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅))) | ||
Theorem | setcthin 45847* | A category of sets all of whose objects contain at most one element is thin. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (SetCat‘𝑈)) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 ∃*𝑝 𝑝 ∈ 𝑥) ⇒ ⊢ (𝜑 → 𝐶 ∈ ThinCat) | ||
Theorem | setc2othin 45848 | The category (SetCat‘2_{o}) is thin. A special case of setcthin 45847. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (SetCat‘2_{o}) ∈ ThinCat | ||
Syntax | cprstc 45849 | Class function defining preordered sets as categories. |
class ProsetToCat | ||
Definition | df-prstc 45850 |
Definition of the function converting a preordered set to a category.
Justified by prsthinc 45846.
This definition is somewhat arbitrary. Example 3.3(4.d) of [Adamek] p. 24 demonstrates an alternate definition with pairwise disjoint hom-sets. The behavior of the function is defined entirely, up to isomorphism, by prstcnid 45853, prstchom 45862, and prstcthin 45861. Other important properties include prstcbas 45854, prstcleval 45855, prstcle 45856, prstcocval 45857, prstcoc 45858, prstchom2 45863, and prstcprs 45860. Use those instead. Note that the defining property prstchom 45862 is equivalent to prstchom2 45863 given prstcthin 45861. See thincn0eu 45836 for justification. "ProsetToCat" was taken instead of "ProsetCat" because the latter might mean the category of preordered sets (classes). However, "ProsetToCat" seems too long. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.) |
⊢ ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1_{o}})⟩) sSet ⟨(comp‘ndx), ∅⟩)) | ||
Theorem | prstcval 45851 | Lemma for prstcnidlem 45852 and prstcthin 45861. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) ⇒ ⊢ (𝜑 → 𝐶 = ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1_{o}})⟩) sSet ⟨(comp‘ndx), ∅⟩)) | ||
Theorem | prstcnidlem 45852 | Lemma for prstcnid 45853 and prstchomval 45859. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝐸‘ndx) ≠ (comp‘ndx) ⇒ ⊢ (𝜑 → (𝐸‘𝐶) = (𝐸‘(𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1_{o}})⟩))) | ||
Theorem | prstcnid 45853 | Components other than Hom and comp are unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝐸‘ndx) ≠ (comp‘ndx) & ⊢ (𝐸‘ndx) ≠ (Hom ‘ndx) ⇒ ⊢ (𝜑 → (𝐸‘𝐾) = (𝐸‘𝐶)) | ||
Theorem | prstcbas 45854 | The base set is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) ⇒ ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) | ||
Theorem | prstcleval 45855 | Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ≤ = (le‘𝐾)) ⇒ ⊢ (𝜑 → ≤ = (le‘𝐶)) | ||
Theorem | prstcle 45856 | Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ≤ = (le‘𝐾)) ⇒ ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ 𝑋(le‘𝐶)𝑌)) | ||
Theorem | prstcocval 45857 | Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ⊥ = (oc‘𝐾)) ⇒ ⊢ (𝜑 → ⊥ = (oc‘𝐶)) | ||
Theorem | prstcoc 45858 | Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ⊥ = (oc‘𝐾)) ⇒ ⊢ (𝜑 → ( ⊥ ‘𝑋) = ((oc‘𝐶)‘𝑋)) | ||
Theorem | prstchomval 45859 | Hom-sets of the constructed category which depend on an arbitrary definition. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ≤ = (le‘𝐶)) ⇒ ⊢ (𝜑 → ( ≤ × {1_{o}}) = (Hom ‘𝐶)) | ||
Theorem | prstcprs 45860 | The category is a preordered set. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) ⇒ ⊢ (𝜑 → 𝐶 ∈ Proset ) | ||
Theorem | prstcthin 45861 | The preordered set is equipped with a thin category. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) ⇒ ⊢ (𝜑 → 𝐶 ∈ ThinCat) | ||
Theorem | prstchom 45862 |
Hom-sets of the constructed category are dependent on the preorder.
Note that prstchom.x and prstchom.y are redundant here due to our definition of ProsetToCat. However, this should not be assumed as it is definition-dependent. Therefore, the two hypotheses are added for explicitness. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ≤ = (le‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) & ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) ⇒ ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅)) | ||
Theorem | prstchom2 45863* |
Hom-sets of the constructed category are dependent on the preorder.
Note that prstchom.x and prstchom.y are redundant here due to our definition of ProsetToCat ( see prstchom2ALT 45864). However, this should not be assumed as it is definition-dependent. Therefore, the two hypotheses are added for explicitness. (Contributed by Zhi Wang, 21-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ≤ = (le‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) & ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) ⇒ ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌))) | ||
Theorem | prstchom2ALT 45864* | Hom-sets of the constructed category are dependent on the preorder. This proof depends on the definition df-prstc 45850. See prstchom2 45863 for a version that does not depend on the definition. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ≤ = (le‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) ⇒ ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌))) | ||
Syntax | cmndtc 45865 | Class function defining monoids as categories. |
class MndToCat | ||
Definition | df-mndtc 45866 |
Definition of the function converting a monoid to a category. Example
3.3(4.e) of [Adamek] p. 24.
The definition of the base set is arbitrary. The whole extensible structure becomes the object here (see mndtcbasval 45868) , instead of just the base set, as is the case in Example 3.3(4.e) of [Adamek] p. 24. The resulting category is defined entirely, up to isomorphism, by mndtcbas 45869, mndtchom 45871, mndtcco 45872. Use those instead. See example 3.26(3) of [Adamek] p. 33 for more on isomorphism. "MndToCat" was taken instead of "MndCat" because the latter might mean the category of monoids. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.) |
⊢ MndToCat = (𝑚 ∈ Mnd ↦ {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+_{g}‘𝑚)⟩}⟩}) | ||
Theorem | mndtcval 45867 | Value of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) ⇒ ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), {𝑀}⟩, ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+_{g}‘𝑀)⟩}⟩}) | ||
Theorem | mndtcbasval 45868 | The base set of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) ⇒ ⊢ (𝜑 → 𝐵 = {𝑀}) | ||
Theorem | mndtcbas 45869* | The category built from a monoid contains precisely one object. (Contributed by Zhi Wang, 22-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) ⇒ ⊢ (𝜑 → ∃!𝑥 𝑥 ∈ 𝐵) | ||
Theorem | mndtcob 45870 | Lemma for mndtchom 45871 and mndtcco 45872. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑋 = 𝑀) | ||
Theorem | mndtchom 45871 | The only hom-set of the category built from a monoid is the base set of the monoid. (Contributed by Zhi Wang, 22-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) ⇒ ⊢ (𝜑 → (𝑋𝐻𝑌) = (Base‘𝑀)) | ||
Theorem | mndtcco 45872 | The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → · = (comp‘𝐶)) ⇒ ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ · 𝑍) = (+_{g}‘𝑀)) | ||
Theorem | mndtcco2 45873 | The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → · = (comp‘𝐶)) & ⊢ (𝜑 → ⚬ = (⟨𝑋, 𝑌⟩ · 𝑍)) ⇒ ⊢ (𝜑 → (𝐺 ⚬ 𝐹) = (𝐺(+_{g}‘𝑀)𝐹)) | ||
Theorem | mndtccatid 45874* | Lemma for mndtccat 45875 and mndtcid 45876. (Contributed by Zhi Wang, 22-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) ⇒ ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ (Base‘𝐶) ↦ (0_{g}‘𝑀)))) | ||
Theorem | mndtccat 45875 | The function value is a category. (Contributed by Zhi Wang, 22-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) ⇒ ⊢ (𝜑 → 𝐶 ∈ Cat) | ||
Theorem | mndtcid 45876 | The identity morphism, or identity arrow, of the category built from a monoid is the identity element of the monoid. (Contributed by Zhi Wang, 22-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 1 = (Id‘𝐶)) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) = (0_{g}‘𝑀)) | ||
Theorem | grptcmon 45877 | All morphisms in a category converted from a group are monomorphisms. (Contributed by Zhi Wang, 23-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝐺)) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → 𝑀 = (Mono‘𝐶)) ⇒ ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌)) | ||
Theorem | grptcepi 45878 | All morphisms in a category converted from a group are epimorphisms. (Contributed by Zhi Wang, 23-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝐺)) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → 𝐸 = (Epi‘𝐶)) ⇒ ⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑋𝐻𝑌)) | ||
Some of these theorems are used in the series of lemmas and theorems proving the defining properties of setrecs. | ||
Theorem | nfintd 45879 | Bound-variable hypothesis builder for intersection. (Contributed by Emmett Weisz, 16-Jan-2020.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥∩ 𝐴) | ||
Theorem | nfiund 45880* | Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.) Add disjoint variable condition to avoid ax-13 2373. See nfiundg 45881 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝐴) & ⊢ (𝜑 → Ⅎ𝑦𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | nfiundg 45881 | Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 2373, see nfiund 45880 for a weaker version that does not require it. (Contributed by Emmett Weisz, 6-Dec-2019.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝐴) & ⊢ (𝜑 → Ⅎ𝑦𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | iunord 45882* | The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. This proof is based on the proof of ssorduni 7531, but does not use it directly, since ssorduni 7531 does not work when 𝐵 is a proper class. (Contributed by Emmett Weisz, 3-Nov-2019.) |
⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → Ord ∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | iunordi 45883* | The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. (Contributed by Emmett Weisz, 3-Nov-2019.) |
⊢ Ord 𝐵 ⇒ ⊢ Ord ∪ 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | spd 45884 | Specialization deduction, using implicit substitution. Based on the proof of spimed 2389. (Contributed by Emmett Weisz, 17-Jan-2020.) |
⊢ (𝜒 → Ⅎ𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝜒 → (∀𝑥𝜑 → 𝜓)) | ||
Theorem | spcdvw 45885* | A version of spcdv 3501 where 𝜓 and 𝜒 are direct substitutions of each other. This theorem is useful because it does not require 𝜑 and 𝑥 to be distinct variables. (Contributed by Emmett Weisz, 12-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) | ||
Theorem | tfis2d 45886* | Transfinite Induction Schema, using implicit substitution. (Contributed by Emmett Weisz, 3-May-2020.) |
⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) & ⊢ (𝜑 → (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜒 → 𝜓))) ⇒ ⊢ (𝜑 → (𝑥 ∈ On → 𝜓)) | ||
Theorem | bnd2d 45887* | Deduction form of bnd2 9407. (Contributed by Emmett Weisz, 19-Jan-2021.) |
⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) ⇒ ⊢ (𝜑 → ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜓)) | ||
Theorem | dffun3f 45888* | Alternate definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Emmett Weisz, 14-Mar-2021.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑧𝐴 ⇒ ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))) | ||
Symbols in this section: All the symbols used in the definition of setrecs(𝐹) are explained in the comment of df-setrecs 45890. The class 𝑌 is explained in the comment of setrec1lem1 45893. Glossaries of symbols used in individual proofs, or used differently in different proofs, are in the comments of those proofs. | ||
Syntax | csetrecs 45889 | Extend class notation to include a set defined by transfinite recursion. |
class setrecs(𝐹) | ||
Definition | df-setrecs 45890* |
Define a class setrecs(𝐹) by transfinite recursion, where
(𝐹‘𝑥) is the set of new elements to add to
the class given the
set 𝑥 of elements in the class so far. We
do not need a base case,
because we can start with the empty set, which is vacuously a subset of
setrecs(𝐹). The goal of this definition is to
construct a class
fulfilling Theorems setrec1 45897 and setrec2v 45902, which give a more
intuitive idea of the meaning of setrecs.
Unlike wrecs,
setrecs is well-defined for any 𝐹 and
meaningful for any
function 𝐹.
For example, see Theorem onsetrec 45913 for how the class On is defined recursively using the successor function. The definition works by building subsets of the desired class and taking the union of those subsets. To find such a collection of subsets, consider an arbitrary set 𝑧, and consider the result when applying 𝐹 to any subset 𝑤 ⊆ 𝑧. Remember that 𝐹 can be any function, and in general we are interested in functions that give outputs that are larger than their inputs, so we have no reason to expect the outputs to be within 𝑧. However, if we restrict the domain of 𝐹 to a given set 𝑦, the resulting range will be a set. Therefore, with this restricted 𝐹, it makes sense to consider sets 𝑧 that are closed under 𝐹 applied to its subsets. Now we can test whether a given set 𝑦 is recursively generated by 𝐹. If every set 𝑧 that is closed under 𝐹 contains 𝑦, that means that every member of 𝑦 must eventually be generated by 𝐹. On the other hand, if some such 𝑧 does not contain a certain element of 𝑦, then that element can be avoided even if we apply 𝐹 in every possible way to previously generated elements. Note that such an omitted element might be eventually recursively generated by 𝐹, but not through the elements of 𝑦. In this case, 𝑦 would fail the condition in the definition, but the omitted element would still be included in some larger 𝑦. For example, if 𝐹 is the successor function, the set {∅, 2_{o}} would fail the condition since 2_{o} is not an element of the successor of ∅ or {∅}. Remember that we are applying 𝐹 to subsets of 𝑦, not elements of 𝑦. In fact, even the set {1_{o}} fails the condition, since the only subset of previously generated elements is ∅, and suc ∅ does not have 1_{o} as an element. However, we can let 𝑦 be any ordinal, since each of its elements is generated by starting with ∅ and repeatedly applying the successor function. A similar definition I initially used for setrecs(𝐹) was setrecs(𝐹) = ∪ ran recs((𝑔 ∈ V ↦ (𝐹‘∪ ran 𝑔))). I had initially tried and failed to find an elementary definition, and I had proven theorems analogous to setrec1 45897 and setrec2v 45902 using the old definition before I found the new one. I decided to change definitions for two reasons. First, as John Horton Conway noted in the Appendix to Part Zero of On Numbers and Games, mathematicians should not be caught up in any particular formalization, such as ZF set theory. Instead, they should work under whatever framework best suits the problem, and the formal bases used for different problems can be shown to be equivalent. Thus, Conway preferred defining surreal numbers as equivalence classes of surreal number forms, rather than sign-expansions. Although sign-expansions are easier to implement in ZF set theory, Conway argued that "formalisation within some particular axiomatic set theory is irrelevant". Furthermore, one of the most remarkable properties of the theory of surreal numbers is that it generates so much from almost nothing. Using sign-expansions as the formal definition destroys the beauty of surreal numbers, because ordinals are already built in. For this reason, I replaced the old definition of setrecs, which also relied heavily on ordinal numbers. On the other hand, both surreal numbers and the elementary definition of setrecs immediately generate the ordinal numbers from a (relatively) very simple set-theoretical basis. Second, although it is still complicated to formalize the theory of recursively generated sets within ZF set theory, it is actually simpler and more natural to do so with set theory directly than with the theory of ordinal numbers. As Conway wrote, indexing the "birthdays" of sets is and should be unnecessary. Using an elementary definition for setrecs removes the reliance on the previously developed theory of ordinal numbers, allowing proofs to be simpler and more direct. Formalizing surreal numbers within Metamath is probably still not in the spirit of Conway. He said that "attempts to force arbitrary theories into a single formal straitjacket... produce unnecessarily cumbrous and inelegant contortions." Nevertheless, Metamath has proven to be much more versatile than it seems at first, and I think the theory of surreal numbers can be natural while fitting well into the Metamath framework. The difficulty in writing a definition in Metamath for setrecs(𝐹) is that the necessary properties to prove are self-referential (see setrec1 45897 and setrec2v 45902), so we cannot simply write the properties we want inside a class abstraction as with most definitions. As noted in the comment of df-rdg 8087, this is not actually a requirement of the Metamath language, but we would like to be able to eliminate all definitions by direct mechanical substitution. We cannot define setrecs using a class abstraction directly, because nothing about its individual elements tells us whether they are in the set. We need to know about previous elements first. One way of getting around this problem without indexing is by defining setrecs(𝐹) as a union or intersection of suitable sets. Thus, instead of using a class abstraction for the elements of setrecs(𝐹), which seems to be impossible, we can use a class abstraction for supersets or subsets of setrecs(𝐹), which "know" about multiple individual elements at a time. Note that we cannot define setrecs(𝐹) as an intersection of sets, because in general it is a proper class, so any supersets would also be proper classes. However, a proper class can be a union of sets, as long as the collection of such sets is a proper class. Therefore, it is feasible to define setrecs(𝐹) as a union of a class abstraction. If setrecs(𝐹) = ∪ 𝐴, the elements of A must be subsets of setrecs(𝐹) which together include everything recursively generated by 𝐹. We can do this by letting 𝐴 be the class of sets 𝑥 whose elements are all recursively generated by 𝐹. One necessary condition is that each element of a given 𝑥 ∈ 𝐴 must be generated by 𝐹 when applied to a previous element 𝑦 ∈ 𝐴. In symbols, ∀𝑥 ∈ 𝐴∃𝑦 ∈ 𝐴(𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐹‘𝑦))}. However, this is not sufficient. All fixed points 𝑥 of 𝐹 will satisfy this condition whether they should be in setrecs(𝐹) or not. If we replace the subset relation with the proper subset relation, 𝑥 cannot be the empty set, even though the empty set should be in 𝐴. Therefore this condition cannot be used in the definition, even if we can find a way to avoid making it circular. A better strategy is to find a necessary and sufficient condition for all the elements of a set 𝑦 ∈ 𝐴 to be generated by 𝐹 when applied only to sets of previously generated elements within 𝑦. For example, taking 𝐹 to be the successor function, we can let 𝐴 = On rather than 𝒫 On, and we will still have ∪ 𝐴 = On as required. This gets rid of the circularity of the definition, since we should have a condition to test whether a given set 𝑦 is in 𝐴 without knowing about any of the other elements of 𝐴. The definition I ended up using accomplishes this using induction: 𝐴 is defined as the class of sets 𝑦 for which a sort of induction on the elements of 𝑦 holds. However, when creating a definition for setrecs that did not rely on ordinal numbers, I tried at first to write a definition using the well-founded relation predicate, Fr. I thought that this would be simple to do once I found a suitable definition using induction, just as the least- element principle is equivalent to induction on the positive integers. If we let 𝑅 = {⟨𝑎, 𝑏⟩ ∣ (𝐹‘𝑎) ⊆ 𝑏}, then (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥∀𝑧 ∈ 𝑥¬ (𝐹‘𝑧) ⊆ 𝑦)). On 22-Jul-2020 I came up with the following definition (Version 1) phrased in terms of induction: ∪ {𝑦 ∣ ∀𝑧 (∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ∈ 𝑧 → (𝐹‘𝑤) ∈ 𝑧)) → 𝑦 ∈ 𝑧)} In Aug-2020 I came up with an equivalent definition with the goal of phrasing it in terms of the relation Fr. It is the contrapositive of the previous one with 𝑧 replaced by its complement. ∪ {𝑦 ∣ ∀𝑧 (𝑦 ∈ 𝑧 → ∃𝑤(𝑤 ⊆ 𝑦 ∧ (𝐹‘𝑤) ∈ 𝑧 ∧ ¬ 𝑤 ∈ 𝑧))} These definitions didn't work because the induction didn't "get off the ground." If 𝑧 does not contain the empty set, the condition (∀𝑤...𝑦 ∈ 𝑧 fails, so 𝑦 = ∅ doesn't get included in 𝐴 even though it should. This could be fixed by adding the base case as a separate requirement, but the subtler problem would remain that rather than a set of "acceptable" sets, what we really need is a collection 𝑧 of all individuals that have been generated so far. So one approach is to replace every occurrence of ∈ 𝑧 with ⊆ 𝑧, making 𝑧 a set of individuals rather than a family of sets. That solves this problem, but it complicates the foundedness version of the definition, which looked cleaner in Version 1. There was another problem with Version 1. If we let 𝐹 be the power set function, then the induction in the inductive version works for 𝑧 being the class of transitive sets, restricted to subsets of 𝑦. Therefore, 𝑦 must be transitive by definition of 𝑧. This doesn't affect the union of all such 𝑦, but it may or may not be desirable. The problem is that 𝐹 is only applied to transitive sets, because of the strong requirement 𝑤 ∈ 𝑧, so the definition requires the additional constraint (𝑎 ⊆ 𝑏 → (𝐹‘𝑎) ⊆ (𝐹‘𝑏)) in order to work. This issue can also be avoided by replacing ∈ 𝑧 with ⊆ 𝑧. The induction version of the result is used in the final definition. Version 2: (18-Aug-2020) Induction: ∪ {𝑦 ∣ ∀𝑧 (∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} Foundedness: ∪ {𝑦 ∣ ∀𝑧(𝑦 ∩ 𝑧 ≠ ∅ → ∃𝑤(𝑤 ⊆ 𝑦 ∧ 𝑤 ∩ 𝑧 = ∅ ∧ (𝐹‘𝑤) ∩ 𝑧 ≠ ∅))} In the induction version, not only does 𝑧 include all the elements of 𝑦, but it must include the elements of (𝐹‘𝑤) for 𝑤 ⊆ (𝑦 ∩ 𝑧) even if those elements of (𝐹‘𝑤) are not in 𝑦. We shouldn't care about any of the elements of 𝑧 outside 𝑦, but this detail doesn't affect the correctness of the definition. If we replaced (𝐹‘𝑤) in the definition by ((𝐹‘𝑤) ∩ 𝑦), we would get the same class for setrecs(𝐹). Suppose we could find a 𝑧 for which the condition fails for a given 𝑦 under the changed definition. Then the antecedent would be true, but 𝑦 ⊆ 𝑧 would be false. We could then simply add all elements of (𝐹‘𝑤) outside of 𝑦 for any 𝑤 ⊆ 𝑦, which we can do because all the classes involved are sets. This is not trivial and requires the axioms of union, power set, and replacement. However, the expanded 𝑧 fails the condition under the Metamath definition. The other direction is easier. If a certain 𝑧 fails the Metamath definition, then all (𝐹‘𝑤) ⊆ 𝑧 for 𝑤 ⊆ (𝑦 ∩ 𝑧), and in particular ((𝐹‘𝑤) ∩ 𝑦) ⊆ 𝑧. The foundedness version is starting to look more like ax-reg 9141! We want to take advantage of the preexisting relation Fr, which seems closely related to our foundedness definition. Since we only care about the elements of 𝑧 which are subsets of 𝑦, we can restrict 𝑧 to 𝑦 in the foundedness definition. Furthermore, instead of quantifying over 𝑤, quantify over the elements 𝑣 ∈ 𝑧 overlapping with 𝑤. Versions 3, 4, and 5 are all equivalent to Version 2. Version 3 - Foundedness (5-Sep-2020): ∪ {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝑦 ∧ 𝑧 ≠ ∅) → ∃𝑣 ∈ 𝑧∃𝑤(𝑤 ⊆ 𝑦 ∧ 𝑤 ∩ 𝑧 = ∅ ∧ 𝑣 ∈ (𝐹‘𝑤)))} Now, if we replace (𝐹‘𝑤) by ((𝐹‘𝑤) ∩ 𝑦), we do not change the definition. We already know that 𝑣 ∈ 𝑦 since 𝑣 ∈ 𝑧 and 𝑧 ⊆ 𝑦. All we need to show in order to prove that this change leads to an equivalent definition is to find To make our definition look exactly like df-fr 5493, we add another variable 𝑢 representing the nonexistent element of 𝑤 in 𝑧. Version 4 - Foundedness (6-Sep-2020): ∪ {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝑦 ∧ 𝑧 ≠ ∅) → ∃𝑣 ∈ 𝑧∃𝑤∀𝑢 ∈ 𝑧(𝑤 ⊆ 𝑦 ∧ ¬ 𝑢 ∈ 𝑤 ∧ 𝑣 ∈ (𝐹‘𝑤)) This is so close to df-fr 5493; the only change needed is to switch ∃𝑤 with ∀𝑢 ∈ 𝑧. Unfortunately, I couldn't find any way to switch the quantifiers without interfering with the definition. Maybe there is a definition equivalent to this one that uses Fr, but I couldn't find one. Yet, we can still find a remarkable similarity between Foundedness Version 2 and ax-reg 9141. Rather than a disjoint element of 𝑧, there's a disjoint coverer of an element of 𝑧. Finally, here's a different dead end I followed: To clean up our foundedness definition, we keep 𝑧 as a family of sets 𝑦 but allow 𝑤 to be any subset of ∪ 𝑧 in the induction. With this stronger induction, we can also allow for the stronger requirement 𝒫 𝑦 ⊆ 𝑧 rather than only 𝑦 ∈ 𝑧. This will help improve the foundedness version. Version 1.1 (28-Aug-2020) Induction: ∪ {𝑦 ∣ ∀𝑧(∀𝑤 (𝑤 ⊆ 𝑦 → (𝑤 ⊆ ∪ 𝑧 → (𝐹‘𝑤) ∈ 𝑧)) → 𝒫 𝑦 ⊆ 𝑧)} Foundedness: ∪ {𝑦 ∣ ∀𝑧(∃𝑎(𝑎 ⊆ 𝑦 ∧ 𝑎 ∈ 𝑧) → ∃𝑤(𝑤 ⊆ 𝑦 ∧ 𝑤 ∩ ∩ 𝑧 = ∅ ∧ (𝐹‘𝑤) ∈ 𝑧))} ( Edit (Aug 31) - this isn't true! Nothing forces the subset of an element of 𝑧 to be in 𝑧. Version 2 does not have this issue. ) Similarly, we could allow 𝑤 to be any subset of any element of 𝑧 rather than any subset of ∪ 𝑧. I think this has the same problem. We want to take advantage of the preexisting relation Fr, which seems closely related to our foundedness definition. Since we only care about the elements of 𝑧 which are subsets of 𝑦, we can restrict 𝑧 to 𝒫 𝑦 in the foundedness definition: Version 1.2 (31-Aug-2020) Foundedness: ∪ {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝒫 𝑦 ∧ 𝑧 ≠ ∅) → ∃𝑤(𝑤 ∈ 𝒫 𝑦 ∧ 𝑤 ∩ ∩ 𝑧 = ∅ ∧ (𝐹‘𝑤) ∈ 𝑧))} Now this looks more like df-fr 5493! The last step necessary to be able to use Fr directly in our definition is to replace (𝐹‘𝑤) with its own setvar variable, corresponding to 𝑦 in df-fr 5493. This definition is incorrect, though, since there's nothing forcing the subset of an element of 𝑧 to be in 𝑧. Version 1.3 (31-Aug-2020) Induction: ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ ∪ 𝑧 → (𝑤 ∈ 𝑧 ∧ (𝐹‘𝑤) ∈ 𝑧))) → 𝒫 𝑦 ⊆ 𝑧)} Foundedness: ∪ {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝒫 𝑦 ∧ 𝑧 ≠ ∅) → ∃𝑤(𝑤 ∈ 𝒫 𝑦 ∧ 𝑤 ∩ ∩ 𝑧 = ∅ ∧ (𝑤 ∈ 𝑧 ∨ (𝐹‘𝑤) ∈ 𝑧)))} 𝑧 must contain the supersets of each of its elements in the foundedness version, and we can't make any restrictions on 𝑧 or 𝐹, so this doesn't work. Let's try letting R be the covering relation 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑏 ∈ (𝐹‘𝑎)} to solve the transitivity issue (i.e. that if 𝐹 is the power set relation, 𝐴 consists only of transitive sets). The set (𝐹‘𝑤) corresponds to the variable 𝑦 in df-fr 5493. Thus, in our case, df-fr 5493 is equivalent to (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅) → ∃𝑤((𝐹‘𝑤) ∈ 𝑧 ∧ ¬ ∃𝑣 ∈ 𝑧𝑣𝑅(𝐹‘𝑤))). Substituting our relation 𝑅 gives (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅) → ∃𝑤((𝐹‘𝑤) ∈ 𝑧 ∧ ¬ ∃𝑣 ∈ 𝑧(𝐹‘𝑤) ∈ (𝐹‘𝑣))) This doesn't work for non-injective 𝐹 because we need all 𝑧 to be straddlers, but we don't necessarily need all-straddlers; loops within z are fine for non-injective F. Consider the foundedness form of Version 1. We want to show ¬ 𝑤 ∈ 𝑧 ↔ ∀𝑣 ∈ 𝑧¬ 𝑣𝑅(𝐹‘𝑤) so we can replace one with the other. Negate both sides: 𝑤 ∈ 𝑧 ↔ ∃𝑣 ∈ 𝑧𝑣𝑅(𝐹‘𝑤) If 𝐹 is injective, then we should be able to pick a suitable R, being careful about the above problem for some F (for example z = transitivity) when changing the antecedent y e. z' to z =/= (/). If we're clever, we can get rid of the injectivity requirement. The forward direction of the above equivalence always holds, but the key is that although the backwards direction doesn't hold in general, we can always find some z' where it doesn't work for 𝑤 itself. If there exists a z' where the version with the w condition fails, then there exists a z' where the version with the v condition also fails. However, Version 1 is not a correct definition, so this doesn't work either. (Contributed by Emmett Weisz, 18-Aug-2020.) (New usage is discouraged.) |
⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} | ||
Theorem | setrecseq 45891 | Equality theorem for set recursion. (Contributed by Emmett Weisz, 17-Feb-2021.) |
⊢ (𝐹 = 𝐺 → setrecs(𝐹) = setrecs(𝐺)) | ||
Theorem | nfsetrecs 45892 | Bound-variable hypothesis builder for setrecs. (Contributed by Emmett Weisz, 21-Oct-2021.) |
⊢ Ⅎ𝑥𝐹 ⇒ ⊢ Ⅎ𝑥setrecs(𝐹) | ||
Theorem | setrec1lem1 45893* |
Lemma for setrec1 45897. This is a utility theorem showing the
equivalence
of the statement 𝑋 ∈ 𝑌 and its expanded form. The proof
uses
elabg 3576 and equivalence theorems.
Variable 𝑌 is the class of sets 𝑦 that are recursively generated by the function 𝐹. In other words, 𝑦 ∈ 𝑌 iff by starting with the empty set and repeatedly applying 𝐹 to subsets 𝑤 of our set, we will eventually generate all the elements of 𝑌. In this theorem, 𝑋 is any element of 𝑌, and 𝑉 is any class. (Contributed by Emmett Weisz, 16-Oct-2020.) (New usage is discouraged.) |
⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝑌 ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧))) | ||
Theorem | setrec1lem2 45894* | Lemma for setrec1 45897. If a family of sets are all recursively generated by 𝐹, so is their union. In this theorem, 𝑋 is a family of sets which are all elements of 𝑌, and 𝑉 is any class. Use dfss3 3875, equivalence and equality theorems, and unissb at the end. Sandwich with applications of setrec1lem1. (Contributed by Emmett Weisz, 24-Jan-2021.) (New usage is discouraged.) |
⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ⊆ 𝑌) ⇒ ⊢ (𝜑 → ∪ 𝑋 ∈ 𝑌) | ||
Theorem | setrec1lem3 45895* | Lemma for setrec1 45897. If each element 𝑎 of 𝐴 is covered by a set 𝑥 recursively generated by 𝐹, then there is a single such set covering all of 𝐴. The set is constructed explicitly using setrec1lem2 45894. It turns out that 𝑥 = 𝐴 also works, i.e., given the hypotheses it is possible to prove that 𝐴 ∈ 𝑌. I don't know if proving this fact directly using setrec1lem1 45893 would be any easier than the current proof using setrec1lem2 45894, and it would only slightly simplify the proof of setrec1 45897. Other than the use of bnd2d 45887, this is a purely technical theorem for rearranging notation from that of setrec1lem2 45894 to that of setrec1 45897. (Contributed by Emmett Weisz, 20-Jan-2021.) (New usage is discouraged.) |
⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝑌)) ⇒ ⊢ (𝜑 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ 𝑌)) | ||
Theorem | setrec1lem4 45896* |
Lemma for setrec1 45897. If 𝑋 is recursively generated by 𝐹, then
so is 𝑋 ∪ (𝐹‘𝐴).
In the proof of setrec1 45897, the following is substituted for this theorem's 𝜑: (𝜑 ∧ (𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤 (𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)})) Therefore, we cannot declare 𝑧 to be a distinct variable from 𝜑, since we need it to appear as a bound variable in 𝜑. This theorem can be proven without the hypothesis Ⅎ𝑧𝜑, but the proof would be harder to read because theorems in deduction form would be interrupted by theorems like eximi 1841, making the antecedent of each line something more complicated than 𝜑. The proof of setrec1lem2 45894 could similarly be made easier to read by adding the hypothesis Ⅎ𝑧𝜑, but I had already finished the proof and decided to leave it as is. (Contributed by Emmett Weisz, 26-Nov-2020.) (New usage is discouraged.) |
⊢ Ⅎ𝑧𝜑 & ⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ (𝜑 → 𝑋 ∈ 𝑌) ⇒ ⊢ (𝜑 → (𝑋 ∪ (𝐹‘𝐴)) ∈ 𝑌) | ||
Theorem | setrec1 45897 |
This is the first of two fundamental theorems about set recursion from
which all other facts will be derived. It states that the class
setrecs(𝐹) is closed under 𝐹. This
effectively sets the
actual value of setrecs(𝐹) as a lower bound for
setrecs(𝐹), as it implies that any set
generated by successive
applications of 𝐹 is a member of 𝐵. This
theorem "gets off the
ground" because we can start by letting 𝐴 = ∅, and the
hypotheses
of the theorem will hold trivially.
Variable 𝐵 represents an abbreviation of setrecs(𝐹) or another name of setrecs(𝐹) (for an example of the latter, see theorem setrecon). Proof summary: Assume that 𝐴 ⊆ 𝐵, meaning that all elements of 𝐴 are in some set recursively generated by 𝐹. Then by setrec1lem3 45895, 𝐴 is a subset of some set recursively generated by 𝐹. (It turns out that 𝐴 itself is recursively generated by 𝐹, but we don't need this fact. See the comment to setrec1lem3 45895.) Therefore, by setrec1lem4 45896, (𝐹‘𝐴) is a subset of some set recursively generated by 𝐹. Thus, by ssuni 4833, it is a subset of the union of all sets recursively generated by 𝐹. See df-setrecs 45890 for a detailed description of how the setrecs definition works. (Contributed by Emmett Weisz, 9-Oct-2020.) |
⊢ 𝐵 = setrecs(𝐹) & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) ⊆ 𝐵) | ||
Theorem | setrec2fun 45898* |
This is the second of two fundamental theorems about set recursion from
which all other facts will be derived. It states that the class
setrecs(𝐹) is a subclass of all classes 𝐶 that
are closed
under 𝐹. Taken together, Theorems setrec1 45897 and setrec2v 45902 say
that setrecs(𝐹) is the minimal class closed under
𝐹.
We express this by saying that if 𝐹 respects the ⊆ relation and 𝐶 is closed under 𝐹, then 𝐵 ⊆ 𝐶. By substituting strategically constructed classes for 𝐶, we can easily prove many useful properties. Although this theorem cannot show equality between 𝐵 and 𝐶, if we intend to prove equality between 𝐵 and some particular class (such as On), we first apply this theorem, then the relevant induction theorem (such as tfi 7599) to the other class. (Contributed by Emmett Weisz, 15-Feb-2021.) (New usage is discouraged.) |
⊢ Ⅎ𝑎𝐹 & ⊢ 𝐵 = setrecs(𝐹) & ⊢ Fun 𝐹 & ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶)) ⇒ ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | ||
Theorem | setrec2lem1 45899* | Lemma for setrec2 45901. The functional part of 𝐹 has the same values as 𝐹. (Contributed by Emmett Weisz, 4-Mar-2021.) (New usage is discouraged.) |
⊢ ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹‘𝑎) | ||
Theorem | setrec2lem2 45900* | Lemma for setrec2 45901. The functional part of 𝐹 is a function. (Contributed by Emmett Weisz, 6-Mar-2021.) (New usage is discouraged.) |
⊢ Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |