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| Mirrors > Home > MPE Home > Th. List > Mathboxes > catprs2 | Structured version Visualization version GIF version | ||
| Description: 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 48875 and catprsc2 48876 for constructions satisfying the hypothesis "catprs.1". See catprs 48873 for a more primitive version. See prsthinc 49084 for constructing a thin category from a proset. (Contributed by Zhi Wang, 18-Sep-2024.) |
| Ref | Expression |
|---|---|
| catprs.1 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅)) |
| catprs.b | ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) |
| catprs.h | ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) |
| catprs.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
| catprs2.l | ⊢ (𝜑 → ≤ = (le‘𝐶)) |
| Ref | Expression |
|---|---|
| catprs2 | ⊢ (𝜑 → 𝐶 ∈ Proset ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | catprs.1 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅)) | |
| 2 | catprs.b | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) | |
| 3 | catprs.h | . . . 4 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) | |
| 4 | catprs.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
| 5 | 1, 2, 3, 4 | catprs 48873 | . . 3 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → (𝑤 ≤ 𝑤 ∧ ((𝑤 ≤ 𝑣 ∧ 𝑣 ≤ 𝑢) → 𝑤 ≤ 𝑢))) |
| 6 | 5 | ralrimivvva 3204 | . 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐵 (𝑤 ≤ 𝑤 ∧ ((𝑤 ≤ 𝑣 ∧ 𝑣 ≤ 𝑢) → 𝑤 ≤ 𝑢))) |
| 7 | catprs2.l | . . 3 ⊢ (𝜑 → ≤ = (le‘𝐶)) | |
| 8 | 2, 7, 4 | isprsd 48825 | . 2 ⊢ (𝜑 → (𝐶 ∈ Proset ↔ ∀𝑤 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐵 (𝑤 ≤ 𝑤 ∧ ((𝑤 ≤ 𝑣 ∧ 𝑣 ≤ 𝑢) → 𝑤 ≤ 𝑢)))) |
| 9 | 6, 8 | mpbird 257 | 1 ⊢ (𝜑 → 𝐶 ∈ Proset ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2939 ∀wral 3060 ∅c0 4332 class class class wbr 5141 ‘cfv 6559 (class class class)co 7429 Basecbs 17243 lecple 17300 Hom chom 17304 Catccat 17703 Proset cproset 18334 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pr 5430 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-cat 17707 df-cid 17708 df-proset 18336 |
| This theorem is referenced by: (None) |
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