Theorem arweuthinc 50637

Description: If a structure has a unique disjointified arrow, then the structure is a thin category. (Contributed by Zhi Wang, 20-Oct-2025.)

Assertion

Ref	Expression
arweuthinc	⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ ThinCat)

Distinct variable group:   𝐶,𝑎

Proof of Theorem arweuthinc

Dummy variables 𝑏 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2761	. 2 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → (Base‘𝐶) = (Base‘𝐶))
2		eqidd 2761	. 2 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → (Hom ‘𝐶) = (Hom ‘𝐶))
3		eqeq1 2764	. . . . . 6 ⊢ (𝑎 = ⟨𝑥, 𝑦, 𝑓⟩ → (𝑎 = 𝑏 ↔ ⟨𝑥, 𝑦, 𝑓⟩ = 𝑏))
4		eqeq2 2772	. . . . . 6 ⊢ (𝑏 = ⟨𝑥, 𝑦, 𝑔⟩ → (⟨𝑥, 𝑦, 𝑓⟩ = 𝑏 ↔ ⟨𝑥, 𝑦, 𝑓⟩ = ⟨𝑥, 𝑦, 𝑔⟩))
5		eumo 2603	. . . . . . . 8 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∃*𝑎 𝑎 ∈ (Arrow‘𝐶))
6	5	ad2antrr 739	. . . . . . 7 ⊢ (((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ∃*𝑎 𝑎 ∈ (Arrow‘𝐶))
7		moel 3385	. . . . . . 7 ⊢ (∃*𝑎 𝑎 ∈ (Arrow‘𝐶) ↔ ∀𝑎 ∈ (Arrow‘𝐶)∀𝑏 ∈ (Arrow‘𝐶)𝑎 = 𝑏)
8	6, 7	sylib 221	. . . . . 6 ⊢ (((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ∀𝑎 ∈ (Arrow‘𝐶)∀𝑏 ∈ (Arrow‘𝐶)𝑎 = 𝑏)
9		eqid 2760	. . . . . . . 8 ⊢ (Arrow‘𝐶) = (Arrow‘𝐶)
10		eqid 2760	. . . . . . . 8 ⊢ (Homa‘𝐶) = (Homa‘𝐶)
11	9, 10	homarw 18221	. . . . . . 7 ⊢ (𝑥(Homa‘𝐶)𝑦) ⊆ (Arrow‘𝐶)
12		eqid 2760	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
13		euex 2602	. . . . . . . . . 10 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∃𝑎 𝑎 ∈ (Arrow‘𝐶))
14	9	arwrcl 18219	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ Cat)
15	14	exlimiv 1962	. . . . . . . . . 10 ⊢ (∃𝑎 𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ Cat)
16	13, 15	syl 18	. . . . . . . . 9 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ Cat)
17	16	ad2antrr 739	. . . . . . . 8 ⊢ (((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝐶 ∈ Cat)
18		eqid 2760	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
19		simplrl 789	. . . . . . . 8 ⊢ (((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑥 ∈ (Base‘𝐶))
20		simplrr 790	. . . . . . . 8 ⊢ (((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑦 ∈ (Base‘𝐶))
21		simprl 783	. . . . . . . 8 ⊢ (((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
22	10, 12, 17, 18, 19, 20, 21	elhomai2 18209	. . . . . . 7 ⊢ (((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ⟨𝑥, 𝑦, 𝑓⟩ ∈ (𝑥(Homa‘𝐶)𝑦))
23	11, 22	sselid 3906	. . . . . 6 ⊢ (((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ⟨𝑥, 𝑦, 𝑓⟩ ∈ (Arrow‘𝐶))
24		simprr 785	. . . . . . . 8 ⊢ (((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))
25	10, 12, 17, 18, 19, 20, 24	elhomai2 18209	. . . . . . 7 ⊢ (((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ⟨𝑥, 𝑦, 𝑔⟩ ∈ (𝑥(Homa‘𝐶)𝑦))
26	11, 25	sselid 3906	. . . . . 6 ⊢ (((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ⟨𝑥, 𝑦, 𝑔⟩ ∈ (Arrow‘𝐶))
27	3, 4, 8, 23, 26	rspc2dv 3590	. . . . 5 ⊢ (((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ⟨𝑥, 𝑦, 𝑓⟩ = ⟨𝑥, 𝑦, 𝑔⟩)
28		vex 3454	. . . . . . 7 ⊢ 𝑥 ∈ V
29		vex 3454	. . . . . . 7 ⊢ 𝑦 ∈ V
30		vex 3454	. . . . . . 7 ⊢ 𝑓 ∈ V
31	28, 29, 30	otth 5452	. . . . . 6 ⊢ (⟨𝑥, 𝑦, 𝑓⟩ = ⟨𝑥, 𝑦, 𝑔⟩ ↔ (𝑥 = 𝑥 ∧ 𝑦 = 𝑦 ∧ 𝑓 = 𝑔))
32	31	simp3bi 1165	. . . . 5 ⊢ (⟨𝑥, 𝑦, 𝑓⟩ = ⟨𝑥, 𝑦, 𝑔⟩ → 𝑓 = 𝑔)
33	27, 32	syl 18	. . . 4 ⊢ (((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑓 = 𝑔)
34	33	ralrimivva 3205	. . 3 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)𝑓 = 𝑔)
35		moel 3385	. . 3 ⊢ (∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)𝑓 = 𝑔)
36	34, 35	sylibr 237	. 2 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
37	1, 2, 36, 16	isthincd 50544	1 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ ThinCat)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∃*wmo 2562  ∃!weu 2593  ∀wral 3076  ⟨cotp 4591  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  Hom chom 17439  Catccat 17838  Arrowcarw 18197  Hom_achoma 18198  ThinCatcthinc 50525

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-ot 4592  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-homa 18201  df-arw 18202  df-thinc 50526

This theorem is used by:  arweutermc  50638