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Theorem prsthinc 47514
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 47471 and catprs2 47472 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 2734 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ( × {1o}) = ( × {1o}))
43f1omo 47367 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ∃*𝑓 𝑓 ∈ (( × {1o})‘⟨𝑥, 𝑦⟩))
5 df-ov 7399 . . . . 5 (𝑥( × {1o})𝑦) = (( × {1o})‘⟨𝑥, 𝑦⟩)
65eleq2i 2826 . . . 4 (𝑓 ∈ (𝑥( × {1o})𝑦) ↔ 𝑓 ∈ (( × {1o})‘⟨𝑥, 𝑦⟩))
76mobii 2543 . . 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 261 . 2 (((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧))) ↔ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧))))
12 0lt1o 8491 . . 3 ∅ ∈ 1o
131eleq2d 2820 . . . . . 6 (𝜑 → (𝑦𝐵𝑦 ∈ (Base‘𝐶)))
14 eqid 2733 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
15 eqid 2733 . . . . . . . 8 (le‘𝐶) = (le‘𝐶)
1614, 15prsref 18239 . . . . . . 7 ((𝐶 ∈ Proset ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦(le‘𝐶)𝑦)
1710, 16sylan 581 . . . . . 6 ((𝜑𝑦 ∈ (Base‘𝐶)) → 𝑦(le‘𝐶)𝑦)
1813, 17sylbida 593 . . . . 5 ((𝜑𝑦𝐵) → 𝑦(le‘𝐶)𝑦)
19 prsthinc.l . . . . . . 7 (𝜑 = (le‘𝐶))
2019breqd 5155 . . . . . 6 (𝜑 → (𝑦 𝑦𝑦(le‘𝐶)𝑦))
2120biimpar 479 . . . . 5 ((𝜑𝑦(le‘𝐶)𝑦) → 𝑦 𝑦)
2218, 21syldan 592 . . . 4 ((𝜑𝑦𝐵) → 𝑦 𝑦)
23 eqidd 2734 . . . . 5 ((𝜑𝑦𝐵) → ( × {1o}) = ( × {1o}))
24 1oex 8463 . . . . . 6 1o ∈ V
2524a1i 11 . . . . 5 ((𝜑𝑦𝐵) → 1o ∈ V)
26 1n0 8475 . . . . . 6 1o ≠ ∅
2726a1i 11 . . . . 5 ((𝜑𝑦𝐵) → 1o ≠ ∅)
2823, 25, 27fvconstr 47362 . . . 4 ((𝜑𝑦𝐵) → (𝑦 𝑦 ↔ (𝑦( × {1o})𝑦) = 1o))
2922, 28mpbid 231 . . 3 ((𝜑𝑦𝐵) → (𝑦( × {1o})𝑦) = 1o)
3012, 29eleqtrrid 2841 . 2 ((𝜑𝑦𝐵) → ∅ ∈ (𝑦( × {1o})𝑦))
31 0ov 7433 . . . . . 6 (⟨𝑥, 𝑦⟩∅𝑧) = ∅
3231oveqi 7409 . . . . 5 (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) = (𝑔𝑓)
33 0ov 7433 . . . . 5 (𝑔𝑓) = ∅
3432, 33eqtri 2761 . . . 4 (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) = ∅
3534, 12eqeltri 2830 . . 3 (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) ∈ 1o
36 simpl 484 . . . . 5 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → 𝜑)
3710adantr 482 . . . . . 6 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → 𝐶 ∈ Proset )
381eleq2d 2820 . . . . . . . . 9 (𝜑 → (𝑥𝐵𝑥 ∈ (Base‘𝐶)))
391eleq2d 2820 . . . . . . . . 9 (𝜑 → (𝑧𝐵𝑧 ∈ (Base‘𝐶)))
4038, 13, 393anbi123d 1437 . . . . . . . 8 (𝜑 → ((𝑥𝐵𝑦𝐵𝑧𝐵) ↔ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))))
4140biimpa 478 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)))
4241adantrr 716 . . . . . 6 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)))
43 eqidd 2734 . . . . . . . 8 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → ( × {1o}) = ( × {1o}))
44 simprrl 780 . . . . . . . 8 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → 𝑓 ∈ (𝑥( × {1o})𝑦))
4543, 44fvconstr2 47364 . . . . . . 7 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → 𝑥 𝑦)
4619breqd 5155 . . . . . . . 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 781 . . . . . . . 8 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → 𝑔 ∈ (𝑦( × {1o})𝑧))
5043, 49fvconstr2 47364 . . . . . . 7 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → 𝑦 𝑧)
5119breqd 5155 . . . . . . . 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 18240 . . . . . 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 1375 . . . . 5 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥( × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( × {1o})𝑧)))) → 𝑥(le‘𝐶)𝑧)
5619breqd 5155 . . . . . 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 47362 . . . 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 2841 . 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 47498 1 (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦𝐵 ↦ ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  ∃*wmo 2533  wne 2941  Vcvv 3475  c0 4320  {csn 4624  cop 4630   class class class wbr 5144  cmpt 5227   × cxp 5670  cfv 6535  (class class class)co 7396  1oc1o 8446  Basecbs 17131  lecple 17191  Hom chom 17195  compcco 17196  Idccid 17596   Proset cproset 18233  ThinCatcthinc 47479
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-riota 7352  df-ov 7399  df-1o 8453  df-cat 17599  df-cid 17600  df-proset 18235  df-thinc 47480
This theorem is referenced by:  prstcthin  47536
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