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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 45910 and catprsc2 45911 for constructions satisfying the hypothesis "catprs.1". See catprs 45908 for a more primitive version. See prsthinc 45951 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 45908 | . . 3 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → (𝑤 ≤ 𝑤 ∧ ((𝑤 ≤ 𝑣 ∧ 𝑣 ≤ 𝑢) → 𝑤 ≤ 𝑢))) |
6 | 5 | ralrimivvva 3103 | . 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 45865 | . 2 ⊢ (𝜑 → (𝐶 ∈ Proset ↔ ∀𝑤 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐵 (𝑤 ≤ 𝑤 ∧ ((𝑤 ≤ 𝑣 ∧ 𝑣 ≤ 𝑢) → 𝑤 ≤ 𝑢)))) |
9 | 6, 8 | mpbird 260 | 1 ⊢ (𝜑 → 𝐶 ∈ Proset ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∀wral 3051 ∅c0 4223 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 lecple 16756 Hom chom 16760 Catccat 17121 Proset cproset 17754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-cat 17125 df-cid 17126 df-proset 17756 |
This theorem is referenced by: (None) |
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