Theorem termcarweu 50866

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 23	. . . 4 ⊢ (𝐶 ∈ TermCat → 𝐶 ∈ TermCat)
2		eqid 2763	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
3	1, 2	termcbas 50818	. . 3 ⊢ (𝐶 ∈ TermCat → ∃𝑥(Base‘𝐶) = {𝑥})
4		eqid 2763	. . . . 5 ⊢ (Homa‘𝐶) = (Homa‘𝐶)
5	1	adantr 486	. . . . . 6 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → 𝐶 ∈ TermCat)
6	5	termccatd 50817	. . . . 5 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → 𝐶 ∈ Cat)
7		eqid 2763	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8		vsnid 4624	. . . . . 6 ⊢ 𝑥 ∈ {𝑥}
9		simpr 490	. . . . . 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 2763	. . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶)
12	2, 7, 11, 6, 10	catidcl 17882	. . . . 5 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
13	4, 2, 6, 7, 10, 10, 12	elhomai2 18235	. . . 4 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ ∈ (𝑥(Homa‘𝐶)𝑥))
14		eqid 2763	. . . . . . . . 9 ⊢ (Arrow‘𝐶) = (Arrow‘𝐶)
15	14	arwdmcd 18253	. . . . . . . 8 ⊢ (𝑎 ∈ (Arrow‘𝐶) → 𝑎 = ⟨(doma‘𝑎), (coda‘𝑎), (2nd ‘𝑎)⟩)
16	15	adantl 487	. . . . . . 7 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → 𝑎 = ⟨(doma‘𝑎), (coda‘𝑎), (2nd ‘𝑎)⟩)
17	5	adantr 486	. . . . . . . . 9 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → 𝐶 ∈ TermCat)
18	14, 2	arwdm 18248	. . . . . . . . . 10 ⊢ (𝑎 ∈ (Arrow‘𝐶) → (doma‘𝑎) ∈ (Base‘𝐶))
19	18	adantl 487	. . . . . . . . 9 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → (doma‘𝑎) ∈ (Base‘𝐶))
20	10	adantr 486	. . . . . . . . 9 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
21	17, 2, 19, 20	termcbasmo 50821	. . . . . . . 8 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → (doma‘𝑎) = 𝑥)
22	14, 2	arwcd 18249	. . . . . . . . . 10 ⊢ (𝑎 ∈ (Arrow‘𝐶) → (coda‘𝑎) ∈ (Base‘𝐶))
23	22	adantl 487	. . . . . . . . 9 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → (coda‘𝑎) ∈ (Base‘𝐶))
24	17, 2, 23, 20	termcbasmo 50821	. . . . . . . 8 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → (coda‘𝑎) = 𝑥)
25	14, 7	arwhom 18252	. . . . . . . . . 10 ⊢ (𝑎 ∈ (Arrow‘𝐶) → (2nd ‘𝑎) ∈ ((doma‘𝑎)(Hom ‘𝐶)(coda‘𝑎)))
26	25	adantl 487	. . . . . . . . 9 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → (2nd ‘𝑎) ∈ ((doma‘𝑎)(Hom ‘𝐶)(coda‘𝑎)))
27	12	adantr 486	. . . . . . . . 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 50823	. . . . . . . 8 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → (2nd ‘𝑎) = ((Id‘𝐶)‘𝑥))
29	21, 24, 28	oteq123d 4869	. . . . . . 7 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → ⟨(doma‘𝑎), (coda‘𝑎), (2nd ‘𝑎)⟩ = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩)
30	16, 29	eqtrd 2798	. . . . . 6 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩)
31		simpr 490	. . . . . . 7 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩) → 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩)
32	14, 4	homarw 18247	. . . . . . . . 9 ⊢ (𝑥(Homa‘𝐶)𝑥) ⊆ (Arrow‘𝐶)
33	32, 13	sselid 3911	. . . . . . . 8 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ ∈ (Arrow‘𝐶))
34	33	adantr 486	. . . . . . 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 813	. . . . 5 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → (𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩))
37	36	alrimiv 1962	. . . 4 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → ∀𝑎(𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩))
38		eqeq2 2775	. . . . . 6 ⊢ (𝑏 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ → (𝑎 = 𝑏 ↔ 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩))
39	38	bibi2d 345	. . . . 5 ⊢ (𝑏 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ → ((𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = 𝑏) ↔ (𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩)))
40	39	albidv 1955	. . . 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 3556	. . 3 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → ∃𝑏∀𝑎(𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = 𝑏))
42	3, 41	exlimddv 1970	. 2 ⊢ (𝐶 ∈ TermCat → ∃𝑏∀𝑎(𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = 𝑏))
43		eu6im 2603	. 2 ⊢ (∃𝑏∀𝑎(𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = 𝑏) → ∃!𝑎 𝑎 ∈ (Arrow‘𝐶))
44	42, 43	syl 18	1 ⊢ (𝐶 ∈ TermCat → ∃!𝑎 𝑎 ∈ (Arrow‘𝐶))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∃!weu 2596  {csn 4588  ⟨cotp 4861  ‘cfv 6555  (class class class)co 7440  2^nd c2nd 8021  Basecbs 17413  Hom chom 17465  Idccid 17865  dom_acdoma 18221  cod_accoda 18222  Arrowcarw 18223  Hom_achoma 18224  TermCatctermc 50810

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-ot 4862  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-1st 8022  df-2nd 8023  df-cat 17868  df-cid 17869  df-doma 18225  df-coda 18226  df-homa 18227  df-arw 18228  df-thinc 50756  df-termc 50811

This theorem is used by:  dftermc3  50869