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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 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.) |
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 45820 | . . 3 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → (𝑤 ≤ 𝑤 ∧ ((𝑤 ≤ 𝑣 ∧ 𝑣 ≤ 𝑢) → 𝑤 ≤ 𝑢))) |
6 | 5 | ralrimivvva 3105 | . 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 45818 | . 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 1542 ∈ wcel 2114 ≠ wne 2935 ∀wral 3054 ∅c0 4221 class class class wbr 5040 ‘cfv 6349 (class class class)co 7182 Basecbs 16598 lecple 16687 Hom chom 16691 Catccat 17050 Proset cproset 17664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pr 5306 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-cat 17054 df-cid 17055 df-proset 17666 |
This theorem is referenced by: (None) |
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