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Theorem catprs2 46293
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 46294 and catprsc2 46295 for constructions satisfying the hypothesis "catprs.1". See catprs 46292 for a more primitive version. See prsthinc 46335 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 46292 . . 3 ((𝜑 ∧ (𝑤𝐵𝑣𝐵𝑢𝐵)) → (𝑤 𝑤 ∧ ((𝑤 𝑣𝑣 𝑢) → 𝑤 𝑢)))
65ralrimivvva 3127 . 2 (𝜑 → ∀𝑤𝐵𝑣𝐵𝑢𝐵 (𝑤 𝑤 ∧ ((𝑤 𝑣𝑣 𝑢) → 𝑤 𝑢)))
7 catprs2.l . . 3 (𝜑 = (le‘𝐶))
82, 7, 4isprsd 46249 . 2 (𝜑 → (𝐶 ∈ Proset ↔ ∀𝑤𝐵𝑣𝐵𝑢𝐵 (𝑤 𝑤 ∧ ((𝑤 𝑣𝑣 𝑢) → 𝑤 𝑢))))
96, 8mpbird 256 1 (𝜑𝐶 ∈ Proset )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wne 2943  wral 3064  c0 4256   class class class wbr 5074  cfv 6433  (class class class)co 7275  Basecbs 16912  lecple 16969  Hom chom 16973  Catccat 17373   Proset cproset 18011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-cat 17377  df-cid 17378  df-proset 18013
This theorem is referenced by: (None)
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