Theorem termcarweu 50154

Description: There exists a unique disjointified arrow in a terminal category. (Contributed by Zhi Wang, 20-Oct-2025.)

Assertion

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

Distinct variable group:   𝐶,𝑎

Proof of Theorem termcarweu

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

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (𝐶 ∈ TermCat → 𝐶 ∈ TermCat)
2		eqid 2764	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
3	1, 2	termcbas 50106	. . 3 ⊢ (𝐶 ∈ TermCat → ∃𝑥(Base‘𝐶) = {𝑥})
4		eqid 2764	. . . . 5 ⊢ (Homa‘𝐶) = (Homa‘𝐶)
5	1	adantr 484	. . . . . 6 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → 𝐶 ∈ TermCat)
6	5	termccd 50105	. . . . 5 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → 𝐶 ∈ Cat)
7		eqid 2764	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8		vsnid 4624	. . . . . 6 ⊢ 𝑥 ∈ {𝑥}
9		simpr 488	. . . . . 6 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → (Base‘𝐶) = {𝑥})
10	8, 9	eleqtrrid 2871	. . . . 5 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → 𝑥 ∈ (Base‘𝐶))
11		eqid 2764	. . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶)
12	2, 7, 11, 6, 10	catidcl 17716	. . . . 5 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
13	4, 2, 6, 7, 10, 10, 12	elhomai2 18069	. . . 4 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ ∈ (𝑥(Homa‘𝐶)𝑥))
14		eqid 2764	. . . . . . . . 9 ⊢ (Arrow‘𝐶) = (Arrow‘𝐶)
15	14	arwdmcd 18087	. . . . . . . 8 ⊢ (𝑎 ∈ (Arrow‘𝐶) → 𝑎 = ⟨(doma‘𝑎), (coda‘𝑎), (2nd ‘𝑎)⟩)
16	15	adantl 485	. . . . . . 7 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → 𝑎 = ⟨(doma‘𝑎), (coda‘𝑎), (2nd ‘𝑎)⟩)
17	5	adantr 484	. . . . . . . . 9 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → 𝐶 ∈ TermCat)
18	14, 2	arwdm 18082	. . . . . . . . . 10 ⊢ (𝑎 ∈ (Arrow‘𝐶) → (doma‘𝑎) ∈ (Base‘𝐶))
19	18	adantl 485	. . . . . . . . 9 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → (doma‘𝑎) ∈ (Base‘𝐶))
20	10	adantr 484	. . . . . . . . 9 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
21	17, 2, 19, 20	termcbasmo 50109	. . . . . . . 8 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → (doma‘𝑎) = 𝑥)
22	14, 2	arwcd 18083	. . . . . . . . . 10 ⊢ (𝑎 ∈ (Arrow‘𝐶) → (coda‘𝑎) ∈ (Base‘𝐶))
23	22	adantl 485	. . . . . . . . 9 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → (coda‘𝑎) ∈ (Base‘𝐶))
24	17, 2, 23, 20	termcbasmo 50109	. . . . . . . 8 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → (coda‘𝑎) = 𝑥)
25	14, 7	arwhom 18086	. . . . . . . . . 10 ⊢ (𝑎 ∈ (Arrow‘𝐶) → (2nd ‘𝑎) ∈ ((doma‘𝑎)(Hom ‘𝐶)(coda‘𝑎)))
26	25	adantl 485	. . . . . . . . 9 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → (2nd ‘𝑎) ∈ ((doma‘𝑎)(Hom ‘𝐶)(coda‘𝑎)))
27	12	adantr 484	. . . . . . . . 9 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
28	17, 2, 19, 23, 7, 26, 20, 20, 27	termchommo 50111	. . . . . . . 8 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → (2nd ‘𝑎) = ((Id‘𝐶)‘𝑥))
29	21, 24, 28	oteq123d 4848	. . . . . . 7 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → ⟨(doma‘𝑎), (coda‘𝑎), (2nd ‘𝑎)⟩ = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩)
30	16, 29	eqtrd 2799	. . . . . 6 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩)
31		simpr 488	. . . . . . 7 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩) → 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩)
32	14, 4	homarw 18081	. . . . . . . . 9 ⊢ (𝑥(Homa‘𝐶)𝑥) ⊆ (Arrow‘𝐶)
33	32, 13	sselid 3936	. . . . . . . 8 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ ∈ (Arrow‘𝐶))
34	33	adantr 484	. . . . . . 7 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ ∈ (Arrow‘𝐶))
35	31, 34	eqeltrd 2864	. . . . . 6 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩) → 𝑎 ∈ (Arrow‘𝐶))
36	30, 35	impbida 810	. . . . 5 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → (𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩))
37	36	alrimiv 1949	. . . 4 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → ∀𝑎(𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩))
38		eqeq2 2776	. . . . . 6 ⊢ (𝑏 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ → (𝑎 = 𝑏 ↔ 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩))
39	38	bibi2d 344	. . . . 5 ⊢ (𝑏 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ → ((𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = 𝑏) ↔ (𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩)))
40	39	albidv 1942	. . . 4 ⊢ (𝑏 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ → (∀𝑎(𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = 𝑏) ↔ ∀𝑎(𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩)))
41	13, 37, 40	spcedv 3559	. . 3 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → ∃𝑏∀𝑎(𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = 𝑏))
42	3, 41	exlimddv 1957	. 2 ⊢ (𝐶 ∈ TermCat → ∃𝑏∀𝑎(𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = 𝑏))
43		eu6im 2604	. 2 ⊢ (∃𝑏∀𝑎(𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = 𝑏) → ∃!𝑎 𝑎 ∈ (Arrow‘𝐶))
44	42, 43	syl 17	1 ⊢ (𝐶 ∈ TermCat → ∃!𝑎 𝑎 ∈ (Arrow‘𝐶))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1560   = wceq 1562  ∃wex 1801   ∈ wcel 2144  ∃!weu 2597  {csn 4584  ⟨cotp 4592  ‘cfv 6523  (class class class)co 7398  2^nd c2nd 7971  Basecbs 17247  Hom chom 17299  Idccid 17699  dom_acdoma 18055  cod_accoda 18056  Arrowcarw 18057  Hom_achoma 18058  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