Theorem arweutermc 50156

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 50155	. 2 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ ThinCat)
2		euex 2606	. . . 4 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∃𝑎 𝑎 ∈ (Arrow‘𝐶))
3		eqid 2764	. . . . . . 7 ⊢ (Arrow‘𝐶) = (Arrow‘𝐶)
4		eqid 2764	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
5	3, 4	arwdm 18082	. . . . . 6 ⊢ (𝑎 ∈ (Arrow‘𝐶) → (doma‘𝑎) ∈ (Base‘𝐶))
6		eleq1 2852	. . . . . 6 ⊢ (𝑥 = (doma‘𝑎) → (𝑥 ∈ (Base‘𝐶) ↔ (doma‘𝑎) ∈ (Base‘𝐶)))
7	5, 5, 6	spcedv 3559	. . . . 5 ⊢ (𝑎 ∈ (Arrow‘𝐶) → ∃𝑥 𝑥 ∈ (Base‘𝐶))
8	7	exlimiv 1952	. . . 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 2768	. . . . . . 7 ⊢ (𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ → (𝑎 = 𝑏 ↔ ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ = 𝑏))
11		eqeq2 2776	. . . . . . 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 2607	. . . . . . . . 9 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∃*𝑎 𝑎 ∈ (Arrow‘𝐶))
13	12	adantr 484	. . . . . . . 8 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ∃*𝑎 𝑎 ∈ (Arrow‘𝐶))
14		moel 3389	. . . . . . . 8 ⊢ (∃*𝑎 𝑎 ∈ (Arrow‘𝐶) ↔ ∀𝑎 ∈ (Arrow‘𝐶)∀𝑏 ∈ (Arrow‘𝐶)𝑎 = 𝑏)
15	13, 14	sylib 220	. . . . . . 7 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ∀𝑎 ∈ (Arrow‘𝐶)∀𝑏 ∈ (Arrow‘𝐶)𝑎 = 𝑏)
16		eqid 2764	. . . . . . . . 9 ⊢ (Homa‘𝐶) = (Homa‘𝐶)
17	3, 16	homarw 18081	. . . . . . . 8 ⊢ (𝑥(Homa‘𝐶)𝑥) ⊆ (Arrow‘𝐶)
18	1	adantr 484	. . . . . . . . . 10 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ ThinCat)
19	18	thinccd 50049	. . . . . . . . 9 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
20		eqid 2764	. . . . . . . . 9 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
21		simprl 780	. . . . . . . . 9 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
22		eqid 2764	. . . . . . . . . 10 ⊢ (Id‘𝐶) = (Id‘𝐶)
23	4, 20, 22, 19, 21	catidcl 17716	. . . . . . . . 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 18069	. . . . . . . 8 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ ∈ (𝑥(Homa‘𝐶)𝑥))
25	17, 24	sselid 3936	. . . . . . 7 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ ∈ (Arrow‘𝐶))
26	3, 16	homarw 18081	. . . . . . . 8 ⊢ (𝑦(Homa‘𝐶)𝑦) ⊆ (Arrow‘𝐶)
27		simprr 782	. . . . . . . . 9 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
28	4, 20, 22, 19, 27	catidcl 17716	. . . . . . . . 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 18069	. . . . . . . 8 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ⟨𝑦, 𝑦, ((Id‘𝐶)‘𝑦)⟩ ∈ (𝑦(Homa‘𝐶)𝑦))
30	26, 29	sselid 3936	. . . . . . 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 3598	. . . . . 6 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ = ⟨𝑦, 𝑦, ((Id‘𝐶)‘𝑦)⟩)
32		vex 3460	. . . . . . . 8 ⊢ 𝑥 ∈ V
33		fvex 6882	. . . . . . . 8 ⊢ ((Id‘𝐶)‘𝑥) ∈ V
34	32, 32, 33	otth 5454	. . . . . . 7 ⊢ (⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ = ⟨𝑦, 𝑦, ((Id‘𝐶)‘𝑦)⟩ ↔ (𝑥 = 𝑦 ∧ 𝑥 = 𝑦 ∧ ((Id‘𝐶)‘𝑥) = ((Id‘𝐶)‘𝑦)))
35	34	simp1bi 1159	. . . . . 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 3207	. . . 4 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)𝑥 = 𝑦)
38		moel 3389	. . . 4 ⊢ (∃*𝑥 𝑥 ∈ (Base‘𝐶) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)𝑥 = 𝑦)
39	37, 38	sylibr 236	. . 3 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∃*𝑥 𝑥 ∈ (Base‘𝐶))
40		df-eu 2598	. . 3 ⊢ (∃!𝑥 𝑥 ∈ (Base‘𝐶) ↔ (∃𝑥 𝑥 ∈ (Base‘𝐶) ∧ ∃*𝑥 𝑥 ∈ (Base‘𝐶)))
41	9, 39, 40	sylanbrc 592	. 2 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∃!𝑥 𝑥 ∈ (Base‘𝐶))
42	4	istermc2 50101	. 2 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃!𝑥 𝑥 ∈ (Base‘𝐶)))
43	1, 41, 42	sylanbrc 592	1 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ TermCat)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562  ∃wex 1801   ∈ wcel 2144  ∃*wmo 2566  ∃!weu 2597  ∀wral 3078  ⟨cotp 4592  ‘cfv 6523  (class class class)co 7398  Basecbs 17247  Hom chom 17299  Idccid 17699  dom_acdoma 18055  Arrowcarw 18057  Hom_achoma 18058  ThinCatcthinc 50043  TermCatctermc 50098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-ot 4593  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-1st 7972  df-2nd 7973  df-cat 17702  df-cid 17703  df-doma 18059  df-coda 18060  df-homa 18061  df-arw 18062  df-thinc 50044  df-termc 50099

This theorem is referenced by:  dftermc3  50157