Theorem arweutermc 49382

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

Assertion

Ref	Expression
arweutermc	⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ TermCat)

Distinct variable group:   𝐶,𝑎

Proof of Theorem arweutermc

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

Step	Hyp	Ref	Expression
1		arweuthinc 49381	. 2 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ ThinCat)
2		euex 2577	. . . 4 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∃𝑎 𝑎 ∈ (Arrow‘𝐶))
3		eqid 2736	. . . . . . 7 ⊢ (Arrow‘𝐶) = (Arrow‘𝐶)
4		eqid 2736	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
5	3, 4	arwdm 18065	. . . . . 6 ⊢ (𝑎 ∈ (Arrow‘𝐶) → (doma‘𝑎) ∈ (Base‘𝐶))
6		eleq1 2823	. . . . . 6 ⊢ (𝑥 = (doma‘𝑎) → (𝑥 ∈ (Base‘𝐶) ↔ (doma‘𝑎) ∈ (Base‘𝐶)))
7	5, 5, 6	spcedv 3582	. . . . 5 ⊢ (𝑎 ∈ (Arrow‘𝐶) → ∃𝑥 𝑥 ∈ (Base‘𝐶))
8	7	exlimiv 1930	. . . 4 ⊢ (∃𝑎 𝑎 ∈ (Arrow‘𝐶) → ∃𝑥 𝑥 ∈ (Base‘𝐶))
9	2, 8	syl 17	. . 3 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∃𝑥 𝑥 ∈ (Base‘𝐶))
10		eqeq1 2740	. . . . . . 7 ⊢ (𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ → (𝑎 = 𝑏 ↔ ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ = 𝑏))
11		eqeq2 2748	. . . . . . 7 ⊢ (𝑏 = ⟨𝑦, 𝑦, ((Id‘𝐶)‘𝑦)⟩ → (⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ = 𝑏 ↔ ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ = ⟨𝑦, 𝑦, ((Id‘𝐶)‘𝑦)⟩))
12		eumo 2578	. . . . . . . . 9 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∃*𝑎 𝑎 ∈ (Arrow‘𝐶))
13	12	adantr 480	. . . . . . . 8 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ∃*𝑎 𝑎 ∈ (Arrow‘𝐶))
14		moel 3386	. . . . . . . 8 ⊢ (∃*𝑎 𝑎 ∈ (Arrow‘𝐶) ↔ ∀𝑎 ∈ (Arrow‘𝐶)∀𝑏 ∈ (Arrow‘𝐶)𝑎 = 𝑏)
15	13, 14	sylib 218	. . . . . . 7 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ∀𝑎 ∈ (Arrow‘𝐶)∀𝑏 ∈ (Arrow‘𝐶)𝑎 = 𝑏)
16		eqid 2736	. . . . . . . . 9 ⊢ (Homa‘𝐶) = (Homa‘𝐶)
17	3, 16	homarw 18064	. . . . . . . 8 ⊢ (𝑥(Homa‘𝐶)𝑥) ⊆ (Arrow‘𝐶)
18	1	adantr 480	. . . . . . . . . 10 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ ThinCat)
19	18	thinccd 49276	. . . . . . . . 9 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
20		eqid 2736	. . . . . . . . 9 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
21		simprl 770	. . . . . . . . 9 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
22		eqid 2736	. . . . . . . . . 10 ⊢ (Id‘𝐶) = (Id‘𝐶)
23	4, 20, 22, 19, 21	catidcl 17699	. . . . . . . . 9 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
24	16, 4, 19, 20, 21, 21, 23	elhomai2 18052	. . . . . . . 8 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ ∈ (𝑥(Homa‘𝐶)𝑥))
25	17, 24	sselid 3961	. . . . . . 7 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ ∈ (Arrow‘𝐶))
26	3, 16	homarw 18064	. . . . . . . 8 ⊢ (𝑦(Homa‘𝐶)𝑦) ⊆ (Arrow‘𝐶)
27		simprr 772	. . . . . . . . 9 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
28	4, 20, 22, 19, 27	catidcl 17699	. . . . . . . . 9 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((Id‘𝐶)‘𝑦) ∈ (𝑦(Hom ‘𝐶)𝑦))
29	16, 4, 19, 20, 27, 27, 28	elhomai2 18052	. . . . . . . 8 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ⟨𝑦, 𝑦, ((Id‘𝐶)‘𝑦)⟩ ∈ (𝑦(Homa‘𝐶)𝑦))
30	26, 29	sselid 3961	. . . . . . 7 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ⟨𝑦, 𝑦, ((Id‘𝐶)‘𝑦)⟩ ∈ (Arrow‘𝐶))
31	10, 11, 15, 25, 30	rspc2dv 3621	. . . . . 6 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ = ⟨𝑦, 𝑦, ((Id‘𝐶)‘𝑦)⟩)
32		vex 3468	. . . . . . . 8 ⊢ 𝑥 ∈ V
33		fvex 6894	. . . . . . . 8 ⊢ ((Id‘𝐶)‘𝑥) ∈ V
34	32, 32, 33	otth 5464	. . . . . . 7 ⊢ (⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ = ⟨𝑦, 𝑦, ((Id‘𝐶)‘𝑦)⟩ ↔ (𝑥 = 𝑦 ∧ 𝑥 = 𝑦 ∧ ((Id‘𝐶)‘𝑥) = ((Id‘𝐶)‘𝑦)))
35	34	simp1bi 1145	. . . . . 6 ⊢ (⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ = ⟨𝑦, 𝑦, ((Id‘𝐶)‘𝑦)⟩ → 𝑥 = 𝑦)
36	31, 35	syl 17	. . . . 5 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 = 𝑦)
37	36	ralrimivva 3188	. . . 4 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)𝑥 = 𝑦)
38		moel 3386	. . . 4 ⊢ (∃*𝑥 𝑥 ∈ (Base‘𝐶) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)𝑥 = 𝑦)
39	37, 38	sylibr 234	. . 3 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∃*𝑥 𝑥 ∈ (Base‘𝐶))
40		df-eu 2569	. . 3 ⊢ (∃!𝑥 𝑥 ∈ (Base‘𝐶) ↔ (∃𝑥 𝑥 ∈ (Base‘𝐶) ∧ ∃*𝑥 𝑥 ∈ (Base‘𝐶)))
41	9, 39, 40	sylanbrc 583	. 2 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∃!𝑥 𝑥 ∈ (Base‘𝐶))
42	4	istermc2 49328	. 2 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃!𝑥 𝑥 ∈ (Base‘𝐶)))
43	1, 41, 42	sylanbrc 583	1 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ TermCat)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃*wmo 2538  ∃!weu 2568  ∀wral 3052  ⟨cotp 4614  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  Hom chom 17287  Idccid 17682  dom_acdoma 18038  Arrowcarw 18040  Hom_achoma 18041  ThinCatcthinc 49270  TermCatctermc 49325

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  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-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-ot 4615  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-1st 7993  df-2nd 7994  df-cat 17685  df-cid 17686  df-doma 18042  df-coda 18043  df-homa 18044  df-arw 18045  df-thinc 49271  df-termc 49326

This theorem is referenced by:  dftermc3  49383