Theorem termcarweu 49533

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 2729	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
3	1, 2	termcbas 49485	. . 3 ⊢ (𝐶 ∈ TermCat → ∃𝑥(Base‘𝐶) = {𝑥})
4		eqid 2729	. . . . 5 ⊢ (Homa‘𝐶) = (Homa‘𝐶)
5	1	adantr 480	. . . . . 6 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → 𝐶 ∈ TermCat)
6	5	termccd 49484	. . . . 5 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → 𝐶 ∈ Cat)
7		eqid 2729	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8		vsnid 4615	. . . . . 6 ⊢ 𝑥 ∈ {𝑥}
9		simpr 484	. . . . . 6 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → (Base‘𝐶) = {𝑥})
10	8, 9	eleqtrrid 2835	. . . . 5 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → 𝑥 ∈ (Base‘𝐶))
11		eqid 2729	. . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶)
12	2, 7, 11, 6, 10	catidcl 17588	. . . . 5 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
13	4, 2, 6, 7, 10, 10, 12	elhomai2 17941	. . . 4 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ ∈ (𝑥(Homa‘𝐶)𝑥))
14		eqid 2729	. . . . . . . . 9 ⊢ (Arrow‘𝐶) = (Arrow‘𝐶)
15	14	arwdmcd 17959	. . . . . . . 8 ⊢ (𝑎 ∈ (Arrow‘𝐶) → 𝑎 = ⟨(doma‘𝑎), (coda‘𝑎), (2nd ‘𝑎)⟩)
16	15	adantl 481	. . . . . . 7 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → 𝑎 = ⟨(doma‘𝑎), (coda‘𝑎), (2nd ‘𝑎)⟩)
17	5	adantr 480	. . . . . . . . 9 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → 𝐶 ∈ TermCat)
18	14, 2	arwdm 17954	. . . . . . . . . 10 ⊢ (𝑎 ∈ (Arrow‘𝐶) → (doma‘𝑎) ∈ (Base‘𝐶))
19	18	adantl 481	. . . . . . . . 9 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → (doma‘𝑎) ∈ (Base‘𝐶))
20	10	adantr 480	. . . . . . . . 9 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
21	17, 2, 19, 20	termcbasmo 49488	. . . . . . . 8 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → (doma‘𝑎) = 𝑥)
22	14, 2	arwcd 17955	. . . . . . . . . 10 ⊢ (𝑎 ∈ (Arrow‘𝐶) → (coda‘𝑎) ∈ (Base‘𝐶))
23	22	adantl 481	. . . . . . . . 9 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → (coda‘𝑎) ∈ (Base‘𝐶))
24	17, 2, 23, 20	termcbasmo 49488	. . . . . . . 8 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → (coda‘𝑎) = 𝑥)
25	14, 7	arwhom 17958	. . . . . . . . . 10 ⊢ (𝑎 ∈ (Arrow‘𝐶) → (2nd ‘𝑎) ∈ ((doma‘𝑎)(Hom ‘𝐶)(coda‘𝑎)))
26	25	adantl 481	. . . . . . . . 9 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → (2nd ‘𝑎) ∈ ((doma‘𝑎)(Hom ‘𝐶)(coda‘𝑎)))
27	12	adantr 480	. . . . . . . . 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 49490	. . . . . . . 8 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → (2nd ‘𝑎) = ((Id‘𝐶)‘𝑥))
29	21, 24, 28	oteq123d 4839	. . . . . . 7 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → ⟨(doma‘𝑎), (coda‘𝑎), (2nd ‘𝑎)⟩ = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩)
30	16, 29	eqtrd 2764	. . . . . 6 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩)
31		simpr 484	. . . . . . 7 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩) → 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩)
32	14, 4	homarw 17953	. . . . . . . . 9 ⊢ (𝑥(Homa‘𝐶)𝑥) ⊆ (Arrow‘𝐶)
33	32, 13	sselid 3933	. . . . . . . 8 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ ∈ (Arrow‘𝐶))
34	33	adantr 480	. . . . . . 7 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ ∈ (Arrow‘𝐶))
35	31, 34	eqeltrd 2828	. . . . . 6 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩) → 𝑎 ∈ (Arrow‘𝐶))
36	30, 35	impbida 800	. . . . 5 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → (𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩))
37	36	alrimiv 1927	. . . 4 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → ∀𝑎(𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩))
38		eqeq2 2741	. . . . . 6 ⊢ (𝑏 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ → (𝑎 = 𝑏 ↔ 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩))
39	38	bibi2d 342	. . . . 5 ⊢ (𝑏 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ → ((𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = 𝑏) ↔ (𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩)))
40	39	albidv 1920	. . . 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 3553	. . 3 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → ∃𝑏∀𝑎(𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = 𝑏))
42	3, 41	exlimddv 1935	. 2 ⊢ (𝐶 ∈ TermCat → ∃𝑏∀𝑎(𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = 𝑏))
43		eu6im 2568	. 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 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃!weu 2561  {csn 4577  ⟨cotp 4585  ‘cfv 6482  (class class class)co 7349  2^nd c2nd 7923  Basecbs 17120  Hom chom 17172  Idccid 17571  dom_acdoma 17927  cod_accoda 17928  Arrowcarw 17929  Hom_achoma 17930  TermCatctermc 49477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-ot 4586  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-1st 7924  df-2nd 7925  df-cat 17574  df-cid 17575  df-doma 17931  df-coda 17932  df-homa 17933  df-arw 17934  df-thinc 49423  df-termc 49478

This theorem is referenced by:  dftermc3  49536