Theorem arweutermc 50025

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 50024	. 2 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ ThinCat)
2		euex 2578	. . . 4 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∃𝑎 𝑎 ∈ (Arrow‘𝐶))
3		eqid 2737	. . . . . . 7 ⊢ (Arrow‘𝐶) = (Arrow‘𝐶)
4		eqid 2737	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
5	3, 4	arwdm 18011	. . . . . 6 ⊢ (𝑎 ∈ (Arrow‘𝐶) → (doma‘𝑎) ∈ (Base‘𝐶))
6		eleq1 2825	. . . . . 6 ⊢ (𝑥 = (doma‘𝑎) → (𝑥 ∈ (Base‘𝐶) ↔ (doma‘𝑎) ∈ (Base‘𝐶)))
7	5, 5, 6	spcedv 3541	. . . . 5 ⊢ (𝑎 ∈ (Arrow‘𝐶) → ∃𝑥 𝑥 ∈ (Base‘𝐶))
8	7	exlimiv 1932	. . . 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 2741	. . . . . . 7 ⊢ (𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ → (𝑎 = 𝑏 ↔ ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ = 𝑏))
11		eqeq2 2749	. . . . . . 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 2579	. . . . . . . . 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 3363	. . . . . . . 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 2737	. . . . . . . . 9 ⊢ (Homa‘𝐶) = (Homa‘𝐶)
17	3, 16	homarw 18010	. . . . . . . 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 49918	. . . . . . . . 9 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
20		eqid 2737	. . . . . . . . 9 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
21		simprl 771	. . . . . . . . 9 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
22		eqid 2737	. . . . . . . . . 10 ⊢ (Id‘𝐶) = (Id‘𝐶)
23	4, 20, 22, 19, 21	catidcl 17645	. . . . . . . . 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 17998	. . . . . . . 8 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ ∈ (𝑥(Homa‘𝐶)𝑥))
25	17, 24	sselid 3920	. . . . . . 7 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ ∈ (Arrow‘𝐶))
26	3, 16	homarw 18010	. . . . . . . 8 ⊢ (𝑦(Homa‘𝐶)𝑦) ⊆ (Arrow‘𝐶)
27		simprr 773	. . . . . . . . 9 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
28	4, 20, 22, 19, 27	catidcl 17645	. . . . . . . . 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 17998	. . . . . . . 8 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ⟨𝑦, 𝑦, ((Id‘𝐶)‘𝑦)⟩ ∈ (𝑦(Homa‘𝐶)𝑦))
30	26, 29	sselid 3920	. . . . . . 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 3580	. . . . . 6 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ = ⟨𝑦, 𝑦, ((Id‘𝐶)‘𝑦)⟩)
32		vex 3434	. . . . . . . 8 ⊢ 𝑥 ∈ V
33		fvex 6851	. . . . . . . 8 ⊢ ((Id‘𝐶)‘𝑥) ∈ V
34	32, 32, 33	otth 5436	. . . . . . 7 ⊢ (⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ = ⟨𝑦, 𝑦, ((Id‘𝐶)‘𝑦)⟩ ↔ (𝑥 = 𝑦 ∧ 𝑥 = 𝑦 ∧ ((Id‘𝐶)‘𝑥) = ((Id‘𝐶)‘𝑦)))
35	34	simp1bi 1146	. . . . . 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 3181	. . . 4 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)𝑥 = 𝑦)
38		moel 3363	. . . 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 2570	. . 3 ⊢ (∃!𝑥 𝑥 ∈ (Base‘𝐶) ↔ (∃𝑥 𝑥 ∈ (Base‘𝐶) ∧ ∃*𝑥 𝑥 ∈ (Base‘𝐶)))
41	9, 39, 40	sylanbrc 584	. 2 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∃!𝑥 𝑥 ∈ (Base‘𝐶))
42	4	istermc2 49970	. 2 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃!𝑥 𝑥 ∈ (Base‘𝐶)))
43	1, 41, 42	sylanbrc 584	1 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ TermCat)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃*wmo 2538  ∃!weu 2569  ∀wral 3052  ⟨cotp 4576  ‘cfv 6496  (class class class)co 7364  Basecbs 17176  Hom chom 17228  Idccid 17628  dom_acdoma 17984  Arrowcarw 17986  Hom_achoma 17987  ThinCatcthinc 49912  TermCatctermc 49967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5306  ax-pr 5374  ax-un 7686

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-ot 4577  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5523  df-xp 5634  df-rel 5635  df-cnv 5636  df-co 5637  df-dm 5638  df-rn 5639  df-res 5640  df-ima 5641  df-iota 6452  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7321  df-ov 7367  df-1st 7939  df-2nd 7940  df-cat 17631  df-cid 17632  df-doma 17988  df-coda 17989  df-homa 17990  df-arw 17991  df-thinc 49913  df-termc 49968

This theorem is referenced by:  dftermc3  50026