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Theorem prsthinc 46335
Description: Preordered sets as categories. Similar to example 3.3(4.d) of [Adamek] p. 24, but the hom-sets are not pairwise disjoint. One can define a functor from the category of prosets to the category of small thin categories. See catprs 46292 and catprs2 46293 for inducing a preorder from a category. Example 3.26(2) of [Adamek] p. 33 indicates that it induces a bijection from the equivalence class of isomorphic small thin categories to the equivalence class of order-isomorphic preordered sets. (Contributed by Zhi Wang, 18-Sep-2024.)
Hypotheses
Ref Expression
indthinc.b (𝜑𝐵 = (Base‘𝐶))
prsthinc.h (𝜑 → ( × {1o}) = (Hom ‘𝐶))
prsthinc.o (𝜑 → ∅ = (comp‘𝐶))
prsthinc.l (𝜑 = (le‘𝐶))
prsthinc.p (𝜑𝐶 ∈ Proset )
Assertion
Ref Expression
prsthinc (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦𝐵 ↦ ∅)))
Distinct variable groups:   𝑦,   𝑦,𝐵   𝑦,𝐶   𝜑,𝑦

Proof of Theorem prsthinc
Dummy variables 𝑓 𝑔 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 indthinc.b . 2 (𝜑𝐵 = (Base‘𝐶))
2 prsthinc.h . 2 (𝜑 → ( × {1o}) = (Hom ‘𝐶))
3 eqidd 2739 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ( × {1o}) = ( × {1o}))
43f1omo 46188 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ∃*𝑓 𝑓 ∈ (( × {1o})‘⟨𝑥, 𝑦⟩))
5 df-ov 7278 . . . . 5 (𝑥( × {1o})𝑦) = (( × {1o})‘⟨𝑥, 𝑦⟩)
65eleq2i 2830 . . . 4 (𝑓 ∈ (𝑥( × {1o})𝑦) ↔ 𝑓 ∈ (( × {1o})‘⟨𝑥, 𝑦⟩))
76mobii 2548 . . 3 (∃*𝑓 𝑓 ∈ (𝑥( × {1o})𝑦) ↔ ∃*𝑓 𝑓 ∈ (( × {1o})‘⟨𝑥, 𝑦⟩))
84, 7sylibr 233 . 2 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥( × {1o})𝑦))
9 prsthinc.o . 2 (𝜑 → ∅ = (comp‘𝐶))
10 prsthinc.p . 2 (𝜑𝐶 ∈ Proset )
11 biid 260 . 2 (((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧))) ↔ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧))))
12 0lt1o 8334 . . 3 ∅ ∈ 1o
131eleq2d 2824 . . . . . 6 (𝜑 → (𝑦𝐵𝑦 ∈ (Base‘𝐶)))
14 eqid 2738 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
15 eqid 2738 . . . . . . . 8 (le‘𝐶) = (le‘𝐶)
1614, 15prsref 18017 . . . . . . 7 ((𝐶 ∈ Proset ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦(le‘𝐶)𝑦)
1710, 16sylan 580 . . . . . 6 ((𝜑𝑦 ∈ (Base‘𝐶)) → 𝑦(le‘𝐶)𝑦)
1813, 17sylbida 592 . . . . 5 ((𝜑𝑦𝐵) → 𝑦(le‘𝐶)𝑦)
19 prsthinc.l . . . . . . 7 (𝜑 = (le‘𝐶))
2019breqd 5085 . . . . . 6 (𝜑 → (𝑦 𝑦𝑦(le‘𝐶)𝑦))
2120biimpar 478 . . . . 5 ((𝜑𝑦(le‘𝐶)𝑦) → 𝑦 𝑦)
2218, 21syldan 591 . . . 4 ((𝜑𝑦𝐵) → 𝑦 𝑦)
23 eqidd 2739 . . . . 5 ((𝜑𝑦𝐵) → ( × {1o}) = ( × {1o}))
24 1oex 8307 . . . . . 6 1o ∈ V
2524a1i 11 . . . . 5 ((𝜑𝑦𝐵) → 1o ∈ V)
26 1n0 8318 . . . . . 6 1o ≠ ∅
2726a1i 11 . . . . 5 ((𝜑𝑦𝐵) → 1o ≠ ∅)
2823, 25, 27fvconstr 46183 . . . 4 ((𝜑𝑦𝐵) → (𝑦 𝑦 ↔ (𝑦( × {1o})𝑦) = 1o))
2922, 28mpbid 231 . . 3 ((𝜑𝑦𝐵) → (𝑦( × {1o})𝑦) = 1o)
3012, 29eleqtrrid 2846 . 2 ((𝜑𝑦𝐵) → ∅ ∈ (𝑦( × {1o})𝑦))
31 0ov 7312 . . . . . 6 (⟨𝑥, 𝑦⟩∅𝑧) = ∅
3231oveqi 7288 . . . . 5 (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) = (𝑔𝑓)
33 0ov 7312 . . . . 5 (𝑔𝑓) = ∅
3432, 33eqtri 2766 . . . 4 (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) = ∅
3534, 12eqeltri 2835 . . 3 (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) ∈ 1o
36 simpl 483 . . . . 5 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → 𝜑)
3710adantr 481 . . . . . 6 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → 𝐶 ∈ Proset )
381eleq2d 2824 . . . . . . . . 9 (𝜑 → (𝑥𝐵𝑥 ∈ (Base‘𝐶)))
391eleq2d 2824 . . . . . . . . 9 (𝜑 → (𝑧𝐵𝑧 ∈ (Base‘𝐶)))
4038, 13, 393anbi123d 1435 . . . . . . . 8 (𝜑 → ((𝑥𝐵𝑦𝐵𝑧𝐵) ↔ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))))
4140biimpa 477 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)))
4241adantrr 714 . . . . . 6 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)))
43 eqidd 2739 . . . . . . . 8 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → ( × {1o}) = ( × {1o}))
44 simprrl 778 . . . . . . . 8 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → 𝑓 ∈ (𝑥( × {1o})𝑦))
4543, 44fvconstr2 46185 . . . . . . 7 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → 𝑥 𝑦)
4619breqd 5085 . . . . . . . 8 (𝜑 → (𝑥 𝑦𝑥(le‘𝐶)𝑦))
4746biimpd 228 . . . . . . 7 (𝜑 → (𝑥 𝑦𝑥(le‘𝐶)𝑦))
4836, 45, 47sylc 65 . . . . . 6 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → 𝑥(le‘𝐶)𝑦)
49 simprrr 779 . . . . . . . 8 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → 𝑔 ∈ (𝑦( × {1o})𝑧))
5043, 49fvconstr2 46185 . . . . . . 7 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → 𝑦 𝑧)
5119breqd 5085 . . . . . . . 8 (𝜑 → (𝑦 𝑧𝑦(le‘𝐶)𝑧))
5251biimpd 228 . . . . . . 7 (𝜑 → (𝑦 𝑧𝑦(le‘𝐶)𝑧))
5336, 50, 52sylc 65 . . . . . 6 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → 𝑦(le‘𝐶)𝑧)
5414, 15prstr 18018 . . . . . 6 ((𝐶 ∈ Proset ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑥(le‘𝐶)𝑦𝑦(le‘𝐶)𝑧)) → 𝑥(le‘𝐶)𝑧)
5537, 42, 48, 53, 54syl112anc 1373 . . . . 5 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → 𝑥(le‘𝐶)𝑧)
5619breqd 5085 . . . . . 6 (𝜑 → (𝑥 𝑧𝑥(le‘𝐶)𝑧))
5756biimprd 247 . . . . 5 (𝜑 → (𝑥(le‘𝐶)𝑧𝑥 𝑧))
5836, 55, 57sylc 65 . . . 4 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → 𝑥 𝑧)
5924a1i 11 . . . . 5 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → 1o ∈ V)
6026a1i 11 . . . . 5 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → 1o ≠ ∅)
6143, 59, 60fvconstr 46183 . . . 4 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → (𝑥 𝑧 ↔ (𝑥( × {1o})𝑧) = 1o))
6258, 61mpbid 231 . . 3 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → (𝑥( × {1o})𝑧) = 1o)
6335, 62eleqtrrid 2846 . 2 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) ∈ (𝑥( × {1o})𝑧))
641, 2, 8, 9, 10, 11, 30, 63isthincd2 46319 1 (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦𝐵 ↦ ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  ∃*wmo 2538  wne 2943  Vcvv 3432  c0 4256  {csn 4561  cop 4567   class class class wbr 5074  cmpt 5157   × cxp 5587  cfv 6433  (class class class)co 7275  1oc1o 8290  Basecbs 16912  lecple 16969  Hom chom 16973  compcco 16974  Idccid 17374   Proset cproset 18011  ThinCatcthinc 46300
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-suc 6272  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-1o 8297  df-cat 17377  df-cid 17378  df-proset 18013  df-thinc 46301
This theorem is referenced by:  prstcthin  46357
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