Theorem termcarweu 49274

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 2734	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
3	1, 2	termcbas 49227	. . 3 ⊢ (𝐶 ∈ TermCat → ∃𝑥(Base‘𝐶) = {𝑥})
4		eqid 2734	. . . . 5 ⊢ (Homa‘𝐶) = (Homa‘𝐶)
5	1	adantr 480	. . . . . 6 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → 𝐶 ∈ TermCat)
6	5	termccd 49226	. . . . 5 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → 𝐶 ∈ Cat)
7		eqid 2734	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8		vsnid 4637	. . . . . 6 ⊢ 𝑥 ∈ {𝑥}
9		simpr 484	. . . . . 6 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → (Base‘𝐶) = {𝑥})
10	8, 9	eleqtrrid 2840	. . . . 5 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → 𝑥 ∈ (Base‘𝐶))
11		eqid 2734	. . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶)
12	2, 7, 11, 6, 10	catidcl 17681	. . . . 5 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
13	4, 2, 6, 7, 10, 10, 12	elhomai2 18034	. . . 4 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩ ∈ (𝑥(Homa‘𝐶)𝑥))
14		eqid 2734	. . . . . . . . 9 ⊢ (Arrow‘𝐶) = (Arrow‘𝐶)
15	14	arwdmcd 18052	. . . . . . . 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 18047	. . . . . . . . . 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 49229	. . . . . . . 8 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → (doma‘𝑎) = 𝑥)
22	14, 2	arwcd 18048	. . . . . . . . . 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 49229	. . . . . . . 8 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → (coda‘𝑎) = 𝑥)
25	14, 7	arwhom 18051	. . . . . . . . . 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 49231	. . . . . . . 8 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → (2nd ‘𝑎) = ((Id‘𝐶)‘𝑥))
29	21, 24, 28	oteq123d 4862	. . . . . . 7 ⊢ (((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) ∧ 𝑎 ∈ (Arrow‘𝐶)) → ⟨(doma‘𝑎), (coda‘𝑎), (2nd ‘𝑎)⟩ = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩)
30	16, 29	eqtrd 2769	. . . . . 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 18046	. . . . . . . . 9 ⊢ (𝑥(Homa‘𝐶)𝑥) ⊆ (Arrow‘𝐶)
33	32, 13	sselid 3954	. . . . . . . 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 2833	. . . . . 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 1926	. . . 4 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → ∀𝑎(𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = ⟨𝑥, 𝑥, ((Id‘𝐶)‘𝑥)⟩))
38		eqeq2 2746	. . . . . 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 1919	. . . 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 3575	. . 3 ⊢ ((𝐶 ∈ TermCat ∧ (Base‘𝐶) = {𝑥}) → ∃𝑏∀𝑎(𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = 𝑏))
42	3, 41	exlimddv 1934	. 2 ⊢ (𝐶 ∈ TermCat → ∃𝑏∀𝑎(𝑎 ∈ (Arrow‘𝐶) ↔ 𝑎 = 𝑏))
43		eu6im 2573	. 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 1537   = wceq 1539  ∃wex 1778   ∈ wcel 2107  ∃!weu 2566  {csn 4599  ⟨cotp 4607  ‘cfv 6528  (class class class)co 7400  2^nd c2nd 7982  Basecbs 17215  Hom chom 17269  Idccid 17664  dom_acdoma 18020  cod_accoda 18021  Arrowcarw 18022  Hom_achoma 18023  TermCatctermc 49219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5247  ax-sep 5264  ax-nul 5274  ax-pow 5333  ax-pr 5400  ax-un 7724

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3357  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-ot 4608  df-uni 4882  df-iun 4967  df-br 5118  df-opab 5180  df-mpt 5200  df-id 5546  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-riota 7357  df-ov 7403  df-1st 7983  df-2nd 7984  df-cat 17667  df-cid 17668  df-doma 18024  df-coda 18025  df-homa 18026  df-arw 18027  df-thinc 49167  df-termc 49220

This theorem is referenced by:  dftermc3  49277