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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 48723 and catprsc2 48724 for constructions satisfying the hypothesis "catprs.1". See catprs 48721 for a more primitive version. See prsthinc 48775 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 48721 | . . 3 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → (𝑤 ≤ 𝑤 ∧ ((𝑤 ≤ 𝑣 ∧ 𝑣 ≤ 𝑢) → 𝑤 ≤ 𝑢))) |
6 | 5 | ralrimivvva 3201 | . 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 48673 | . 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 1535 ∈ wcel 2104 ≠ wne 2936 ∀wral 3057 ∅c0 4339 class class class wbr 5149 ‘cfv 6558 (class class class)co 7425 Basecbs 17234 lecple 17294 Hom chom 17298 Catccat 17698 Proset cproset 18339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5430 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 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 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7381 df-ov 7428 df-cat 17702 df-cid 17703 df-proset 18341 |
This theorem is referenced by: (None) |
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