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Theorem catprs2 48722
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.)
Hypotheses
Ref Expression
catprs.1 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))
catprs.b (𝜑𝐵 = (Base‘𝐶))
catprs.h (𝜑𝐻 = (Hom ‘𝐶))
catprs.c (𝜑𝐶 ∈ Cat)
catprs2.l (𝜑 = (le‘𝐶))
Assertion
Ref Expression
catprs2 (𝜑𝐶 ∈ Proset )
Distinct variable groups:   𝑥, ,𝑦   𝑥,𝐵,𝑦   𝑥,𝐻,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem catprs2
Dummy variables 𝑤 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catprs.1 . . . 4 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))
2 catprs.b . . . 4 (𝜑𝐵 = (Base‘𝐶))
3 catprs.h . . . 4 (𝜑𝐻 = (Hom ‘𝐶))
4 catprs.c . . . 4 (𝜑𝐶 ∈ Cat)
51, 2, 3, 4catprs 48721 . . 3 ((𝜑 ∧ (𝑤𝐵𝑣𝐵𝑢𝐵)) → (𝑤 𝑤 ∧ ((𝑤 𝑣𝑣 𝑢) → 𝑤 𝑢)))
65ralrimivvva 3201 . 2 (𝜑 → ∀𝑤𝐵𝑣𝐵𝑢𝐵 (𝑤 𝑤 ∧ ((𝑤 𝑣𝑣 𝑢) → 𝑤 𝑢)))
7 catprs2.l . . 3 (𝜑 = (le‘𝐶))
82, 7, 4isprsd 48673 . 2 (𝜑 → (𝐶 ∈ Proset ↔ ∀𝑤𝐵𝑣𝐵𝑢𝐵 (𝑤 𝑤 ∧ ((𝑤 𝑣𝑣 𝑢) → 𝑤 𝑢))))
96, 8mpbird 257 1 (𝜑𝐶 ∈ 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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