Theorem indthinc 49087

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 49089, 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	⊢ (𝜑 → 𝐵 = (Base‘𝐶))
indthinc.h	⊢ (𝜑 → ((𝐵 × 𝐵) × {1o}) = (Hom ‘𝐶))
indthinc.o	⊢ (𝜑 → ∅ = (comp‘𝐶))
indthinc.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)

Assertion

Ref	Expression
indthinc	⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅)))

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.

Step	Hyp	Ref	Expression
1		indthinc.b	. 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
2		indthinc.h	. 2 ⊢ (𝜑 → ((𝐵 × 𝐵) × {1o}) = (Hom ‘𝐶))
3		eqidd 2735	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐵 × 𝐵) × {1o}) = ((𝐵 × 𝐵) × {1o}))
4	3	f1omo 48751	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (((𝐵 × 𝐵) × {1o})‘⟨𝑥, 𝑦⟩))
5		df-ov 7416	. . . . 5 ⊢ (𝑥((𝐵 × 𝐵) × {1o})𝑦) = (((𝐵 × 𝐵) × {1o})‘⟨𝑥, 𝑦⟩)
6	5	eleq2i 2825	. . . 4 ⊢ (𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ↔ 𝑓 ∈ (((𝐵 × 𝐵) × {1o})‘⟨𝑥, 𝑦⟩))
7	6	mobii 2546	. . 3 ⊢ (∃*𝑓 𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ↔ ∃*𝑓 𝑓 ∈ (((𝐵 × 𝐵) × {1o})‘⟨𝑥, 𝑦⟩))
8	4, 7	sylibr 234	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦))
9		indthinc.o	. 2 ⊢ (𝜑 → ∅ = (comp‘𝐶))
10		indthinc.c	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
11		biid 261	. 2 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ∧ 𝑔 ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑧))) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ∧ 𝑔 ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑧))))
12		id 22	. . . . 5 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐵)
13	12	ancli 548	. . . 4 ⊢ (𝑦 ∈ 𝐵 → (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
14		1oex 8498	. . . . 5 ⊢ 1o ∈ V
15	14	ovconst2 7595	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑦((𝐵 × 𝐵) × {1o})𝑦) = 1o)
16		0lt1o 8524	. . . . 5 ⊢ ∅ ∈ 1o
17		eleq2 2822	. . . . 5 ⊢ ((𝑦((𝐵 × 𝐵) × {1o})𝑦) = 1o → (∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦) ↔ ∅ ∈ 1o))
18	16, 17	mpbiri 258	. . . 4 ⊢ ((𝑦((𝐵 × 𝐵) × {1o})𝑦) = 1o → ∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦))
19	13, 15, 18	3syl 18	. . 3 ⊢ (𝑦 ∈ 𝐵 → ∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦))
20	19	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦))
21	16	a1i 11	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ∅ ∈ 1o)
22		0ov 7450	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩∅𝑧) = ∅
23	22	oveqi 7426	. . . . . 6 ⊢ (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) = (𝑔∅𝑓)
24		0ov 7450	. . . . . 6 ⊢ (𝑔∅𝑓) = ∅
25	23, 24	eqtri 2757	. . . . 5 ⊢ (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) = ∅
26	25	a1i 11	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) = ∅)
27	14	ovconst2 7595	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑧) = 1o)
28	27	3adant2 1131	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑧) = 1o)
29	21, 26, 28	3eltr4d 2848	. . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑧))
30	29	ad2antrl 728	. 2 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ∧ 𝑔 ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑧)))) → (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑧))
31	1, 2, 8, 9, 10, 11, 20, 30	isthincd2 49064	1 ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∃*wmo 2536  ∅c0 4313  {csn 4606  ⟨cop 4612   ↦ cmpt 5205   × cxp 5663  ‘cfv 6541  (class class class)co 7413  1_oc1o 8481  Basecbs 17229  Hom chom 17284  compcco 17285  Idccid 17679  ThinCatcthinc 49044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-1o 8488  df-cat 17682  df-cid 17683  df-thinc 49045

This theorem is referenced by: (None)