Users' Mathboxes Mathbox for Zhi Wang < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  indthinc Structured version   Visualization version   GIF version

Theorem indthinc 48853
Description: An indiscrete category in which all hom-sets have exactly one morphism is a thin category. Constructed here is an indiscrete category where all morphisms are . This is a special case of prsthinc 48855, where = (𝐵 × 𝐵). This theorem also implies a functor from the category of sets to the category of small categories. (Contributed by Zhi Wang, 17-Sep-2024.) (Proof shortened by Zhi Wang, 19-Sep-2024.)
Hypotheses
Ref Expression
indthinc.b (𝜑𝐵 = (Base‘𝐶))
indthinc.h (𝜑 → ((𝐵 × 𝐵) × {1o}) = (Hom ‘𝐶))
indthinc.o (𝜑 → ∅ = (comp‘𝐶))
indthinc.c (𝜑𝐶𝑉)
Assertion
Ref Expression
indthinc (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦𝐵 ↦ ∅)))
Distinct variable groups:   𝑦,𝐵   𝑦,𝐶   𝜑,𝑦
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem indthinc
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 indthinc.h . 2 (𝜑 → ((𝐵 × 𝐵) × {1o}) = (Hom ‘𝐶))
3 eqidd 2736 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝐵 × 𝐵) × {1o}) = ((𝐵 × 𝐵) × {1o}))
43f1omo 48691 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ∃*𝑓 𝑓 ∈ (((𝐵 × 𝐵) × {1o})‘⟨𝑥, 𝑦⟩))
5 df-ov 7434 . . . . 5 (𝑥((𝐵 × 𝐵) × {1o})𝑦) = (((𝐵 × 𝐵) × {1o})‘⟨𝑥, 𝑦⟩)
65eleq2i 2831 . . . 4 (𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ↔ 𝑓 ∈ (((𝐵 × 𝐵) × {1o})‘⟨𝑥, 𝑦⟩))
76mobii 2546 . . 3 (∃*𝑓 𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ↔ ∃*𝑓 𝑓 ∈ (((𝐵 × 𝐵) × {1o})‘⟨𝑥, 𝑦⟩))
84, 7sylibr 234 . 2 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦))
9 indthinc.o . 2 (𝜑 → ∅ = (comp‘𝐶))
10 indthinc.c . 2 (𝜑𝐶𝑉)
11 biid 261 . 2 (((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ∧ 𝑔 ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑧))) ↔ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ∧ 𝑔 ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑧))))
12 id 22 . . . . 5 (𝑦𝐵𝑦𝐵)
1312ancli 548 . . . 4 (𝑦𝐵 → (𝑦𝐵𝑦𝐵))
14 1oex 8515 . . . . 5 1o ∈ V
1514ovconst2 7613 . . . 4 ((𝑦𝐵𝑦𝐵) → (𝑦((𝐵 × 𝐵) × {1o})𝑦) = 1o)
16 0lt1o 8541 . . . . 5 ∅ ∈ 1o
17 eleq2 2828 . . . . 5 ((𝑦((𝐵 × 𝐵) × {1o})𝑦) = 1o → (∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦) ↔ ∅ ∈ 1o))
1816, 17mpbiri 258 . . . 4 ((𝑦((𝐵 × 𝐵) × {1o})𝑦) = 1o → ∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦))
1913, 15, 183syl 18 . . 3 (𝑦𝐵 → ∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦))
2019adantl 481 . 2 ((𝜑𝑦𝐵) → ∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦))
2116a1i 11 . . . 4 ((𝑥𝐵𝑦𝐵𝑧𝐵) → ∅ ∈ 1o)
22 0ov 7468 . . . . . . 7 (⟨𝑥, 𝑦⟩∅𝑧) = ∅
2322oveqi 7444 . . . . . 6 (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) = (𝑔𝑓)
24 0ov 7468 . . . . . 6 (𝑔𝑓) = ∅
2523, 24eqtri 2763 . . . . 5 (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) = ∅
2625a1i 11 . . . 4 ((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) = ∅)
2714ovconst2 7613 . . . . 5 ((𝑥𝐵𝑧𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑧) = 1o)
28273adant2 1130 . . . 4 ((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑧) = 1o)
2921, 26, 283eltr4d 2854 . . 3 ((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑧))
3029ad2antrl 728 . 2 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ∧ 𝑔 ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑧)))) → (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑧))
311, 2, 8, 9, 10, 11, 20, 30isthincd2 48838 1 (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦𝐵 ↦ ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  ∃*wmo 2536  c0 4339  {csn 4631  cop 4637  cmpt 5231   × cxp 5687  cfv 6563  (class class class)co 7431  1oc1o 8498  Basecbs 17245  Hom chom 17309  compcco 17310  Idccid 17710  ThinCatcthinc 48819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-1o 8505  df-cat 17713  df-cid 17714  df-thinc 48820
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator