Theorem arweutermc 50032

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 50031	. 2 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ ThinCat)
2		euex 2583	. . . 4 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∃𝑎 𝑎 ∈ (Arrow‘𝐶))
3		eqid 2741	. . . . . . 7 ⊢ (Arrow‘𝐶) = (Arrow‘𝐶)
4		eqid 2741	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
5	3, 4	arwdm 18009	. . . . . 6 ⊢ (𝑎 ∈ (Arrow‘𝐶) → (doma‘𝑎) ∈ (Base‘𝐶))
6		eleq1 2829	. . . . . 6 ⊢ (𝑥 = (doma‘𝑎) → (𝑥 ∈ (Base‘𝐶) ↔ (doma‘𝑎) ∈ (Base‘𝐶)))
7	5, 5, 6	spcedv 3537	. . . . 5 ⊢ (𝑎 ∈ (Arrow‘𝐶) → ∃𝑥 𝑥 ∈ (Base‘𝐶))
8	7	exlimiv 1938	. . . 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 2745	. . . . . . 7 ⊢ (𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ → (𝑎 = 𝑏 ↔ ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ = 𝑏))
11		eqeq2 2753	. . . . . . 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 2584	. . . . . . . . 9 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∃*𝑎 𝑎 ∈ (Arrow‘𝐶))
13	12	adantr 482	. . . . . . . 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 220	. . . . . . 7 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ∀𝑎 ∈ (Arrow‘𝐶)∀𝑏 ∈ (Arrow‘𝐶)𝑎 = 𝑏)
16		eqid 2741	. . . . . . . . 9 ⊢ (Homa‘𝐶) = (Homa‘𝐶)
17	3, 16	homarw 18008	. . . . . . . 8 ⊢ (𝑥(Homa‘𝐶)𝑥) ⊆ (Arrow‘𝐶)
18	1	adantr 482	. . . . . . . . . 10 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ ThinCat)
19	18	thinccd 49925	. . . . . . . . 9 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
20		eqid 2741	. . . . . . . . 9 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
21		simprl 777	. . . . . . . . 9 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
22		eqid 2741	. . . . . . . . . 10 ⊢ (Id‘𝐶) = (Id‘𝐶)
23	4, 20, 22, 19, 21	catidcl 17643	. . . . . . . . 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 17996	. . . . . . . 8 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ ∈ (𝑥(Homa‘𝐶)𝑥))
25	17, 24	sselid 3914	. . . . . . 7 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ ∈ (Arrow‘𝐶))
26	3, 16	homarw 18008	. . . . . . . 8 ⊢ (𝑦(Homa‘𝐶)𝑦) ⊆ (Arrow‘𝐶)
27		simprr 779	. . . . . . . . 9 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
28	4, 20, 22, 19, 27	catidcl 17643	. . . . . . . . 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 17996	. . . . . . . 8 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ⟨𝑦, 𝑦, ((Id‘𝐶)‘𝑦)⟩ ∈ (𝑦(Homa‘𝐶)𝑦))
30	26, 29	sselid 3914	. . . . . . 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 3576	. . . . . 6 ⊢ ((∃!𝑎 𝑎 ∈ (Arrow‘𝐶) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ = ⟨𝑦, 𝑦, ((Id‘𝐶)‘𝑦)⟩)
32		vex 3437	. . . . . . . 8 ⊢ 𝑥 ∈ V
33		fvex 6843	. . . . . . . 8 ⊢ ((Id‘𝐶)‘𝑥) ∈ V
34	32, 32, 33	otth 5426	. . . . . . 7 ⊢ (⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ = ⟨𝑦, 𝑦, ((Id‘𝐶)‘𝑦)⟩ ↔ (𝑥 = 𝑦 ∧ 𝑥 = 𝑦 ∧ ((Id‘𝐶)‘𝑥) = ((Id‘𝐶)‘𝑦)))
35	34	simp1bi 1152	. . . . . 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 3184	. . . 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 236	. . 3 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∃*𝑥 𝑥 ∈ (Base‘𝐶))
40		df-eu 2575	. . 3 ⊢ (∃!𝑥 𝑥 ∈ (Base‘𝐶) ↔ (∃𝑥 𝑥 ∈ (Base‘𝐶) ∧ ∃*𝑥 𝑥 ∈ (Base‘𝐶)))
41	9, 39, 40	sylanbrc 590	. 2 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → ∃!𝑥 𝑥 ∈ (Base‘𝐶))
42	4	istermc2 49977	. 2 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃!𝑥 𝑥 ∈ (Base‘𝐶)))
43	1, 41, 42	sylanbrc 590	1 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ TermCat)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548  ∃wex 1787   ∈ wcel 2121  ∃*wmo 2543  ∃!weu 2574  ∀wral 3055  ⟨cotp 4565  ‘cfv 6488  (class class class)co 7359  Basecbs 17174  Hom chom 17226  Idccid 17626  dom_acdoma 17982  Arrowcarw 17984  Hom_achoma 17985  ThinCatcthinc 49919  TermCatctermc 49974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-ot 4566  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7316  df-ov 7362  df-1st 7933  df-2nd 7934  df-cat 17629  df-cid 17630  df-doma 17986  df-coda 17987  df-homa 17988  df-arw 17989  df-thinc 49920  df-termc 49975

This theorem is referenced by:  dftermc3  50033