Theorem indthinc 50281

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 50283, 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 2767	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐵 × 𝐵) × {1o}) = ((𝐵 × 𝐵) × {1o}))
4	3	f1omo 49712	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (((𝐵 × 𝐵) × {1o})‘⟨𝑥, 𝑦⟩))
5		df-ov 7426	. . . . 5 ⊢ (𝑥((𝐵 × 𝐵) × {1o})𝑦) = (((𝐵 × 𝐵) × {1o})‘⟨𝑥, 𝑦⟩)
6	5	eleq2i 2858	. . . 4 ⊢ (𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ↔ 𝑓 ∈ (((𝐵 × 𝐵) × {1o})‘⟨𝑥, 𝑦⟩))
7	6	mobii 2579	. . 3 ⊢ (∃*𝑓 𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ↔ ∃*𝑓 𝑓 ∈ (((𝐵 × 𝐵) × {1o})‘⟨𝑥, 𝑦⟩))
8	4, 7	sylibr 237	. 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 264	. 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 23	. . . . 5 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐵)
13	12	ancli 558	. . . 4 ⊢ (𝑦 ∈ 𝐵 → (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
14		1oex 8472	. . . . 5 ⊢ 1o ∈ V
15	14	ovconst2 7603	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑦((𝐵 × 𝐵) × {1o})𝑦) = 1o)
16		0lt1o 8498	. . . . 5 ⊢ ∅ ∈ 1o
17		eleq2 2855	. . . . 5 ⊢ ((𝑦((𝐵 × 𝐵) × {1o})𝑦) = 1o → (∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦) ↔ ∅ ∈ 1o))
18	16, 17	mpbiri 261	. . . 4 ⊢ ((𝑦((𝐵 × 𝐵) × {1o})𝑦) = 1o → ∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦))
19	13, 15, 18	3syl 19	. . 3 ⊢ (𝑦 ∈ 𝐵 → ∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦))
20	19	adantl 487	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦))
21	16	a1i 11	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ∅ ∈ 1o)
22		0ov 7460	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩∅𝑧) = ∅
23	22	oveqi 7436	. . . . . 6 ⊢ (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) = (𝑔∅𝑓)
24		0ov 7460	. . . . . 6 ⊢ (𝑔∅𝑓) = ∅
25	23, 24	eqtri 2789	. . . . 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 7603	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑧) = 1o)
28	27	3adant2 1149	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑧) = 1o)
29	21, 26, 28	3eltr4d 2881	. . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑧))
30	29	ad2antrl 741	. 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 50256	1 ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∃*wmo 2568  ∅c0 4289  {csn 4594  ⟨cop 4600   ↦ cmpt 5197   × cxp 5664  ‘cfv 6543  (class class class)co 7423  1_oc1o 8455  Basecbs 17294  Hom chom 17346  compcco 17347  Idccid 17746  ThinCatcthinc 50236

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-1o 8462  df-cat 17749  df-cid 17750  df-thinc 50237

This theorem is used by: (None)