Theorem arweuthinc 49534

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 2730	. 2 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → (Base‘𝐶) = (Base‘𝐶))
2		eqidd 2730	. 2 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → (Hom ‘𝐶) = (Hom ‘𝐶))
3		eqeq1 2733	. . . . . 6 ⊢ (𝑎 = ⟨𝑥, 𝑦, 𝑓⟩ → (𝑎 = 𝑏 ↔ ⟨𝑥, 𝑦, 𝑓⟩ = 𝑏))
4		eqeq2 2741	. . . . . 6 ⊢ (𝑏 = ⟨𝑥, 𝑦, 𝑔⟩ → (⟨𝑥, 𝑦, 𝑓⟩ = 𝑏 ↔ ⟨𝑥, 𝑦, 𝑓⟩ = ⟨𝑥, 𝑦, 𝑔⟩))
5		eumo 2571	. . . . . . . 8 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∃*𝑎 𝑎 ∈ (Arrow‘𝐶))
6	5	ad2antrr 726	. . . . . . 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 3365	. . . . . . 7 ⊢ (∃*𝑎 𝑎 ∈ (Arrow‘𝐶) ↔ ∀𝑎 ∈ (Arrow‘𝐶)∀𝑏 ∈ (Arrow‘𝐶)𝑎 = 𝑏)
8	6, 7	sylib 218	. . . . . 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 2729	. . . . . . . 8 ⊢ (Arrow‘𝐶) = (Arrow‘𝐶)
10		eqid 2729	. . . . . . . 8 ⊢ (Homa‘𝐶) = (Homa‘𝐶)
11	9, 10	homarw 17953	. . . . . . 7 ⊢ (𝑥(Homa‘𝐶)𝑦) ⊆ (Arrow‘𝐶)
12		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
13		euex 2570	. . . . . . . . . 10 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∃𝑎 𝑎 ∈ (Arrow‘𝐶))
14	9	arwrcl 17951	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ Cat)
15	14	exlimiv 1930	. . . . . . . . . 10 ⊢ (∃𝑎 𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ Cat)
16	13, 15	syl 17	. . . . . . . . 9 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ Cat)
17	16	ad2antrr 726	. . . . . . . 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 2729	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
19		simplrl 776	. . . . . . . 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 777	. . . . . . . 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 770	. . . . . . . 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 17941	. . . . . . 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 3933	. . . . . 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 772	. . . . . . . 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 17941	. . . . . . 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 3933	. . . . . 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 3592	. . . . 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 3440	. . . . . . 7 ⊢ 𝑥 ∈ V
29		vex 3440	. . . . . . 7 ⊢ 𝑦 ∈ V
30		vex 3440	. . . . . . 7 ⊢ 𝑓 ∈ V
31	28, 29, 30	otth 5427	. . . . . 6 ⊢ (⟨𝑥, 𝑦, 𝑓⟩ = ⟨𝑥, 𝑦, 𝑔⟩ ↔ (𝑥 = 𝑥 ∧ 𝑦 = 𝑦 ∧ 𝑓 = 𝑔))
32	31	simp3bi 1147	. . . . 5 ⊢ (⟨𝑥, 𝑦, 𝑓⟩ = ⟨𝑥, 𝑦, 𝑔⟩ → 𝑓 = 𝑔)
33	27, 32	syl 17	. . . 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 3172	. . 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 3365	. . 3 ⊢ (∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)𝑓 = 𝑔)
36	34, 35	sylibr 234	. 2 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
37	1, 2, 36, 16	isthincd 49441	1 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ ThinCat)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃*wmo 2531  ∃!weu 2561  ∀wral 3044  ⟨cotp 4585  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  Hom chom 17172  Catccat 17570  Arrowcarw 17929  Hom_achoma 17930  ThinCatcthinc 49422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-ot 4586  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-homa 17933  df-arw 17934  df-thinc 49423

This theorem is referenced by:  arweutermc  49535