Theorem arweuthinc 49275

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 2735	. 2 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → (Base‘𝐶) = (Base‘𝐶))
2		eqidd 2735	. 2 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → (Hom ‘𝐶) = (Hom ‘𝐶))
3		eqeq1 2738	. . . . . 6 ⊢ (𝑎 = ⟨𝑥, 𝑦, 𝑓⟩ → (𝑎 = 𝑏 ↔ ⟨𝑥, 𝑦, 𝑓⟩ = 𝑏))
4		eqeq2 2746	. . . . . 6 ⊢ (𝑏 = ⟨𝑥, 𝑦, 𝑔⟩ → (⟨𝑥, 𝑦, 𝑓⟩ = 𝑏 ↔ ⟨𝑥, 𝑦, 𝑓⟩ = ⟨𝑥, 𝑦, 𝑔⟩))
5		eumo 2576	. . . . . . . 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 3379	. . . . . . 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 2734	. . . . . . . 8 ⊢ (Arrow‘𝐶) = (Arrow‘𝐶)
10		eqid 2734	. . . . . . . 8 ⊢ (Homa‘𝐶) = (Homa‘𝐶)
11	9, 10	homarw 18046	. . . . . . 7 ⊢ (𝑥(Homa‘𝐶)𝑦) ⊆ (Arrow‘𝐶)
12		eqid 2734	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
13		euex 2575	. . . . . . . . . 10 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∃𝑎 𝑎 ∈ (Arrow‘𝐶))
14	9	arwrcl 18044	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ Cat)
15	14	exlimiv 1929	. . . . . . . . . 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 2734	. . . . . . . 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 18034	. . . . . . 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 3954	. . . . . 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 18034	. . . . . . 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 3954	. . . . . 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 3614	. . . . 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 3461	. . . . . . 7 ⊢ 𝑥 ∈ V
29		vex 3461	. . . . . . 7 ⊢ 𝑦 ∈ V
30		vex 3461	. . . . . . 7 ⊢ 𝑓 ∈ V
31	28, 29, 30	otth 5457	. . . . . 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 3185	. . 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 3379	. . 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 49185	1 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ ThinCat)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107  ∃*wmo 2536  ∃!weu 2566  ∀wral 3050  ⟨cotp 4607  ‘cfv 6528  (class class class)co 7400  Basecbs 17215  Hom chom 17269  Catccat 17663  Arrowcarw 18022  Hom_achoma 18023  ThinCatcthinc 49166

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 5247  ax-sep 5264  ax-nul 5274  ax-pow 5333  ax-pr 5400  ax-un 7724

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-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-ot 4608  df-uni 4882  df-iun 4967  df-br 5118  df-opab 5180  df-mpt 5200  df-id 5546  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7403  df-homa 18026  df-arw 18027  df-thinc 49167

This theorem is referenced by:  arweutermc  49276