Theorem arweutermc 49630

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 49629	. 2 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ ThinCat)
2		euex 2572	. . . 4 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∃𝑎 𝑎 ∈ (Arrow‘𝐶))
3		eqid 2731	. . . . . . 7 ⊢ (Arrow‘𝐶) = (Arrow‘𝐶)
4		eqid 2731	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
5	3, 4	arwdm 17954	. . . . . 6 ⊢ (𝑎 ∈ (Arrow‘𝐶) → (doma‘𝑎) ∈ (Base‘𝐶))
6		eleq1 2819	. . . . . 6 ⊢ (𝑥 = (doma‘𝑎) → (𝑥 ∈ (Base‘𝐶) ↔ (doma‘𝑎) ∈ (Base‘𝐶)))
7	5, 5, 6	spcedv 3548	. . . . 5 ⊢ (𝑎 ∈ (Arrow‘𝐶) → ∃𝑥 𝑥 ∈ (Base‘𝐶))
8	7	exlimiv 1931	. . . 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 2735	. . . . . . 7 ⊢ (𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ → (𝑎 = 𝑏 ↔ ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ = 𝑏))
11		eqeq2 2743	. . . . . . 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 2573	. . . . . . . . 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 3366	. . . . . . . 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 2731	. . . . . . . . 9 ⊢ (Homa‘𝐶) = (Homa‘𝐶)
17	3, 16	homarw 17953	. . . . . . . 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 49523	. . . . . . . . 9 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
20		eqid 2731	. . . . . . . . 9 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
21		simprl 770	. . . . . . . . 9 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
22		eqid 2731	. . . . . . . . . 10 ⊢ (Id‘𝐶) = (Id‘𝐶)
23	4, 20, 22, 19, 21	catidcl 17588	. . . . . . . . 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 17941	. . . . . . . 8 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ ∈ (𝑥(Homa‘𝐶)𝑥))
25	17, 24	sselid 3927	. . . . . . 7 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ ∈ (Arrow‘𝐶))
26	3, 16	homarw 17953	. . . . . . . 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 17588	. . . . . . . . 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 17941	. . . . . . . 8 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ⟨𝑦, 𝑦, ((Id‘𝐶)‘𝑦)⟩ ∈ (𝑦(Homa‘𝐶)𝑦))
30	26, 29	sselid 3927	. . . . . . 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 3587	. . . . . 6 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ = ⟨𝑦, 𝑦, ((Id‘𝐶)‘𝑦)⟩)
32		vex 3440	. . . . . . . 8 ⊢ 𝑥 ∈ V
33		fvex 6835	. . . . . . . 8 ⊢ ((Id‘𝐶)‘𝑥) ∈ V
34	32, 32, 33	otth 5422	. . . . . . 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 3175	. . . 4 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)𝑥 = 𝑦)
38		moel 3366	. . . 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 2564	. . 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 49575	. 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 1541  ∃wex 1780   ∈ wcel 2111  ∃*wmo 2533  ∃!weu 2563  ∀wral 3047  ⟨cotp 4581  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  Hom chom 17172  Idccid 17571  dom_acdoma 17927  Arrowcarw 17929  Hom_achoma 17930  ThinCatcthinc 49517  TermCatctermc 49572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-ot 4582  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-1st 7921  df-2nd 7922  df-cat 17574  df-cid 17575  df-doma 17931  df-coda 17932  df-homa 17933  df-arw 17934  df-thinc 49518  df-termc 49573

This theorem is referenced by:  dftermc3  49631