Theorem termcarweu 49514

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 49466	. . 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 49465	. . . . 5 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → 𝐶 ∈ Cat)
7		eqid 2729	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8		vsnid 4627	. . . . . 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 17643	. . . . 5 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
13	4, 2, 6, 7, 10, 10, 12	elhomai2 17996	. . . 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 18014	. . . . . . . 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 18009	. . . . . . . . . 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 49469	. . . . . . . 8 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → (doma‘𝑎) = 𝑥)
22	14, 2	arwcd 18010	. . . . . . . . . 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 49469	. . . . . . . 8 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → (coda‘𝑎) = 𝑥)
25	14, 7	arwhom 18013	. . . . . . . . . 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 49471	. . . . . . . 8 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → (2nd ‘𝑎) = ((Id‘𝐶)‘𝑥))
29	21, 24, 28	oteq123d 4852	. . . . . . 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 18008	. . . . . . . . 9 ⊢ (𝑥(Homa‘𝐶)𝑥) ⊆ (Arrow‘𝐶)
33	32, 13	sselid 3944	. . . . . . . 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 3564	. . 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 4589  ⟨cotp 4597  ‘cfv 6511  (class class class)co 7387  2^nd c2nd 7967  Basecbs 17179  Hom chom 17231  Idccid 17626  dom_acdoma 17982  cod_accoda 17983  Arrowcarw 17984  Hom_achoma 17985  TermCatctermc 49458

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-ot 4598  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-1st 7968  df-2nd 7969  df-cat 17629  df-cid 17630  df-doma 17986  df-coda 17987  df-homa 17988  df-arw 17989  df-thinc 49404  df-termc 49459

This theorem is referenced by:  dftermc3  49517