Theorem termc2 50296

Description: If there exists a unique functor from both the category itself and the trivial category, then the category is terminal. Note that the converse also holds, so that it is a biconditional. See the proof of termc 50297 for hints. See also eufunc 50300 and euendfunc2 50305 for some insights on why two categories are sufficient. (Contributed by Zhi Wang, 18-Oct-2025.) (Proof shortened by Zhi Wang, 20-Oct-2025.)

Assertion

Ref	Expression
termc2	⊢ (∀𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)∃!𝑓 𝑓 ∈ (𝑑 Func 𝐶) → 𝐶 ∈ TermCat)

Distinct variable group:   𝐶,𝑑,𝑓

Proof of Theorem termc2

Step	Hyp	Ref	Expression
1		eqid 2763	. 2 ⊢ (CatCat‘{𝐶, (SetCat‘1o)}) = (CatCat‘{𝐶, (SetCat‘1o)})
2		fvex 6894	. . . . . 6 ⊢ (SetCat‘1o) ∈ V
3	2	prid2 4729	. . . . 5 ⊢ (SetCat‘1o) ∈ {𝐶, (SetCat‘1o)}
4		setc1oterm 50269	. . . . 5 ⊢ (SetCat‘1o) ∈ TermCat
5	3, 4	elini 4152	. . . 4 ⊢ (SetCat‘1o) ∈ ({𝐶, (SetCat‘1o)} ∩ TermCat)
6	5	ne0ii 4297	. . 3 ⊢ ({𝐶, (SetCat‘1o)} ∩ TermCat) ≠ ∅
7	6	a1i 11	. 2 ⊢ (∀𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)∃!𝑓 𝑓 ∈ (𝑑 Func 𝐶) → ({𝐶, (SetCat‘1o)} ∩ TermCat) ≠ ∅)
8	4	a1i 11	. . . . . . . . 9 ⊢ (⊤ → (SetCat‘1o) ∈ TermCat)
9	8	termccd 50257	. . . . . . . 8 ⊢ (⊤ → (SetCat‘1o) ∈ Cat)
10	9	mptru 1577	. . . . . . 7 ⊢ (SetCat‘1o) ∈ Cat
11	3, 10	elini 4152	. . . . . 6 ⊢ (SetCat‘1o) ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)
12		oveq1 7417	. . . . . . . . 9 ⊢ (𝑑 = (SetCat‘1o) → (𝑑 Func 𝐶) = ((SetCat‘1o) Func 𝐶))
13	12	eleq2d 2849	. . . . . . . 8 ⊢ (𝑑 = (SetCat‘1o) → (𝑓 ∈ (𝑑 Func 𝐶) ↔ 𝑓 ∈ ((SetCat‘1o) Func 𝐶)))
14	13	eubidv 2614	. . . . . . 7 ⊢ (𝑑 = (SetCat‘1o) → (∃!𝑓 𝑓 ∈ (𝑑 Func 𝐶) ↔ ∃!𝑓 𝑓 ∈ ((SetCat‘1o) Func 𝐶)))
15	14	rspcv 3577	. . . . . 6 ⊢ ((SetCat‘1o) ∈ ({𝐶, (SetCat‘1o)} ∩ Cat) → (∀𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)∃!𝑓 𝑓 ∈ (𝑑 Func 𝐶) → ∃!𝑓 𝑓 ∈ ((SetCat‘1o) Func 𝐶)))
16	11, 15	ax-mp 5	. . . . 5 ⊢ (∀𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)∃!𝑓 𝑓 ∈ (𝑑 Func 𝐶) → ∃!𝑓 𝑓 ∈ ((SetCat‘1o) Func 𝐶))
17		euen1b 9021	. . . . 5 ⊢ (((SetCat‘1o) Func 𝐶) ≈ 1o ↔ ∃!𝑓 𝑓 ∈ ((SetCat‘1o) Func 𝐶))
18	16, 17	sylibr 237	. . . 4 ⊢ (∀𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)∃!𝑓 𝑓 ∈ (𝑑 Func 𝐶) → ((SetCat‘1o) Func 𝐶) ≈ 1o)
19		eqid 2763	. . . . . . . . 9 ⊢ (Base‘(CatCat‘{𝐶, (SetCat‘1o)})) = (Base‘(CatCat‘{𝐶, (SetCat‘1o)}))
20		prex 5409	. . . . . . . . . 10 ⊢ {𝐶, (SetCat‘1o)} ∈ V
21	20	a1i 11	. . . . . . . . 9 ⊢ (⊤ → {𝐶, (SetCat‘1o)} ∈ V)
22	1, 19, 21	catcbas 18153	. . . . . . . 8 ⊢ (⊤ → (Base‘(CatCat‘{𝐶, (SetCat‘1o)})) = ({𝐶, (SetCat‘1o)} ∩ Cat))
23	22	mptru 1577	. . . . . . 7 ⊢ (Base‘(CatCat‘{𝐶, (SetCat‘1o)})) = ({𝐶, (SetCat‘1o)} ∩ Cat)
24	23	eqcomi 2772	. . . . . 6 ⊢ ({𝐶, (SetCat‘1o)} ∩ Cat) = (Base‘(CatCat‘{𝐶, (SetCat‘1o)}))
25		eqid 2763	. . . . . 6 ⊢ (Hom ‘(CatCat‘{𝐶, (SetCat‘1o)})) = (Hom ‘(CatCat‘{𝐶, (SetCat‘1o)}))
26	1	catccat 18160	. . . . . . . 8 ⊢ ({𝐶, (SetCat‘1o)} ∈ V → (CatCat‘{𝐶, (SetCat‘1o)}) ∈ Cat)
27	20, 26	ax-mp 5	. . . . . . 7 ⊢ (CatCat‘{𝐶, (SetCat‘1o)}) ∈ Cat
28	27	a1i 11	. . . . . 6 ⊢ (((SetCat‘1o) Func 𝐶) ≈ 1o → (CatCat‘{𝐶, (SetCat‘1o)}) ∈ Cat)
29		euex 2605	. . . . . . . . . 10 ⊢ (∃!𝑓 𝑓 ∈ ((SetCat‘1o) Func 𝐶) → ∃𝑓 𝑓 ∈ ((SetCat‘1o) Func 𝐶))
30		funcrcl 17915	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ ((SetCat‘1o) Func 𝐶) → ((SetCat‘1o) ∈ Cat ∧ 𝐶 ∈ Cat))
31	30	simprd 500	. . . . . . . . . . 11 ⊢ (𝑓 ∈ ((SetCat‘1o) Func 𝐶) → 𝐶 ∈ Cat)
32	31	exlimiv 1960	. . . . . . . . . 10 ⊢ (∃𝑓 𝑓 ∈ ((SetCat‘1o) Func 𝐶) → 𝐶 ∈ Cat)
33	29, 32	syl 18	. . . . . . . . 9 ⊢ (∃!𝑓 𝑓 ∈ ((SetCat‘1o) Func 𝐶) → 𝐶 ∈ Cat)
34	17, 33	sylbi 220	. . . . . . . 8 ⊢ (((SetCat‘1o) Func 𝐶) ≈ 1o → 𝐶 ∈ Cat)
35		prid1g 4726	. . . . . . . 8 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ {𝐶, (SetCat‘1o)})
36	34, 35	syl 18	. . . . . . 7 ⊢ (((SetCat‘1o) Func 𝐶) ≈ 1o → 𝐶 ∈ {𝐶, (SetCat‘1o)})
37	36, 34	elind 4153	. . . . . 6 ⊢ (((SetCat‘1o) Func 𝐶) ≈ 1o → 𝐶 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat))
38	24, 25, 28, 37	istermo 18049	. . . . 5 ⊢ (((SetCat‘1o) Func 𝐶) ≈ 1o → (𝐶 ∈ (TermO‘(CatCat‘{𝐶, (SetCat‘1o)})) ↔ ∀𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)∃!𝑓 𝑓 ∈ (𝑑(Hom ‘(CatCat‘{𝐶, (SetCat‘1o)}))𝐶)))
39	20	a1i 11	. . . . . . . . 9 ⊢ ((((SetCat‘1o) Func 𝐶) ≈ 1o ∧ 𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)) → {𝐶, (SetCat‘1o)} ∈ V)
40		simpr 489	. . . . . . . . 9 ⊢ ((((SetCat‘1o) Func 𝐶) ≈ 1o ∧ 𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)) → 𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat))
41	37	adantr 485	. . . . . . . . 9 ⊢ ((((SetCat‘1o) Func 𝐶) ≈ 1o ∧ 𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)) → 𝐶 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat))
42	1, 24, 39, 25, 40, 41	catchom 18155	. . . . . . . 8 ⊢ ((((SetCat‘1o) Func 𝐶) ≈ 1o ∧ 𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)) → (𝑑(Hom ‘(CatCat‘{𝐶, (SetCat‘1o)}))𝐶) = (𝑑 Func 𝐶))
43	42	eleq2d 2849	. . . . . . 7 ⊢ ((((SetCat‘1o) Func 𝐶) ≈ 1o ∧ 𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)) → (𝑓 ∈ (𝑑(Hom ‘(CatCat‘{𝐶, (SetCat‘1o)}))𝐶) ↔ 𝑓 ∈ (𝑑 Func 𝐶)))
44	43	eubidv 2614	. . . . . 6 ⊢ ((((SetCat‘1o) Func 𝐶) ≈ 1o ∧ 𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)) → (∃!𝑓 𝑓 ∈ (𝑑(Hom ‘(CatCat‘{𝐶, (SetCat‘1o)}))𝐶) ↔ ∃!𝑓 𝑓 ∈ (𝑑 Func 𝐶)))
45	44	ralbidva 3186	. . . . 5 ⊢ (((SetCat‘1o) Func 𝐶) ≈ 1o → (∀𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)∃!𝑓 𝑓 ∈ (𝑑(Hom ‘(CatCat‘{𝐶, (SetCat‘1o)}))𝐶) ↔ ∀𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)∃!𝑓 𝑓 ∈ (𝑑 Func 𝐶)))
46	38, 45	bitrd 282	. . . 4 ⊢ (((SetCat‘1o) Func 𝐶) ≈ 1o → (𝐶 ∈ (TermO‘(CatCat‘{𝐶, (SetCat‘1o)})) ↔ ∀𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)∃!𝑓 𝑓 ∈ (𝑑 Func 𝐶)))
47	18, 46	syl 18	. . 3 ⊢ (∀𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)∃!𝑓 𝑓 ∈ (𝑑 Func 𝐶) → (𝐶 ∈ (TermO‘(CatCat‘{𝐶, (SetCat‘1o)})) ↔ ∀𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)∃!𝑓 𝑓 ∈ (𝑑 Func 𝐶)))
48	47	ibir 271	. 2 ⊢ (∀𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)∃!𝑓 𝑓 ∈ (𝑑 Func 𝐶) → 𝐶 ∈ (TermO‘(CatCat‘{𝐶, (SetCat‘1o)})))
49	1, 7, 48	termcterm2 50292	1 ⊢ (∀𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)∃!𝑓 𝑓 ∈ (𝑑 Func 𝐶) → 𝐶 ∈ TermCat)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570  ⊤wtru 1571  ∃wex 1809   ∈ wcel 2143  ∃!weu 2596   ≠ wne 2958  ∀wral 3079  Vcvv 3455   ∩ cin 3904  ∅c0 4286  {cpr 4591   class class class wbr 5109  ‘cfv 6536  (class class class)co 7410  1_oc1o 8442   ≈ cen 8936  Basecbs 17264  Hom chom 17316  Catccat 17715   Func cfunc 17906  TermOctermo 18034  SetCatcsetc 18127  CatCatccatc 18150  TermCatctermc 50250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-supp 8153  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-z 12587  df-dec 12707  df-uz 12858  df-fz 13531  df-struct 17202  df-slot 17237  df-ndx 17249  df-base 17265  df-hom 17329  df-cco 17330  df-cat 17719  df-cid 17720  df-sect 17799  df-inv 17800  df-iso 17801  df-cic 17848  df-func 17910  df-idfu 17911  df-cofu 17912  df-full 17958  df-fth 17959  df-termo 18037  df-setc 18128  df-catc 18151  df-thinc 50196  df-termc 50251

This theorem is referenced by:  termc  50297