Theorem termc2 50180

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 50181 for hints. See also eufunc 50184 and euendfunc2 50189 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 2769	. 2 ⊢ (CatCat‘{𝐶, (SetCat‘1o)}) = (CatCat‘{𝐶, (SetCat‘1o)})
2		fvex 6895	. . . . . 6 ⊢ (SetCat‘1o) ∈ V
3	2	prid2 4734	. . . . 5 ⊢ (SetCat‘1o) ∈ {𝐶, (SetCat‘1o)}
4		setc1oterm 50153	. . . . 5 ⊢ (SetCat‘1o) ∈ TermCat
5	3, 4	elini 4160	. . . 4 ⊢ (SetCat‘1o) ∈ ({𝐶, (SetCat‘1o)} ∩ TermCat)
6	5	ne0ii 4305	. . 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 50141	. . . . . . . 8 ⊢ (⊤ → (SetCat‘1o) ∈ Cat)
10	9	mptru 1574	. . . . . . 7 ⊢ (SetCat‘1o) ∈ Cat
11	3, 10	elini 4160	. . . . . 6 ⊢ (SetCat‘1o) ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)
12		oveq1 7418	. . . . . . . . 9 ⊢ (𝑑 = (SetCat‘1o) → (𝑑 Func 𝐶) = ((SetCat‘1o) Func 𝐶))
13	12	eleq2d 2855	. . . . . . . 8 ⊢ (𝑑 = (SetCat‘1o) → (𝑓 ∈ (𝑑 Func 𝐶) ↔ 𝑓 ∈ ((SetCat‘1o) Func 𝐶)))
14	13	eubidv 2620	. . . . . . 7 ⊢ (𝑑 = (SetCat‘1o) → (∃!𝑓 𝑓 ∈ (𝑑 Func 𝐶) ↔ ∃!𝑓 𝑓 ∈ ((SetCat‘1o) Func 𝐶)))
15	14	rspcv 3586	. . . . . 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 9024	. . . . 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 2769	. . . . . . . . 9 ⊢ (Base‘(CatCat‘{𝐶, (SetCat‘1o)})) = (Base‘(CatCat‘{𝐶, (SetCat‘1o)}))
20		prex 5410	. . . . . . . . . 10 ⊢ {𝐶, (SetCat‘1o)} ∈ V
21	20	a1i 11	. . . . . . . . 9 ⊢ (⊤ → {𝐶, (SetCat‘1o)} ∈ V)
22	1, 19, 21	catcbas 18157	. . . . . . . 8 ⊢ (⊤ → (Base‘(CatCat‘{𝐶, (SetCat‘1o)})) = ({𝐶, (SetCat‘1o)} ∩ Cat))
23	22	mptru 1574	. . . . . . 7 ⊢ (Base‘(CatCat‘{𝐶, (SetCat‘1o)})) = ({𝐶, (SetCat‘1o)} ∩ Cat)
24	23	eqcomi 2778	. . . . . 6 ⊢ ({𝐶, (SetCat‘1o)} ∩ Cat) = (Base‘(CatCat‘{𝐶, (SetCat‘1o)}))
25		eqid 2769	. . . . . 6 ⊢ (Hom ‘(CatCat‘{𝐶, (SetCat‘1o)})) = (Hom ‘(CatCat‘{𝐶, (SetCat‘1o)}))
26	1	catccat 18164	. . . . . . . 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 2611	. . . . . . . . . 10 ⊢ (∃!𝑓 𝑓 ∈ ((SetCat‘1o) Func 𝐶) → ∃𝑓 𝑓 ∈ ((SetCat‘1o) Func 𝐶))
30		funcrcl 17919	. . . . . . . . . . . 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 1957	. . . . . . . . . 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 4731	. . . . . . . 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 4161	. . . . . 6 ⊢ (((SetCat‘1o) Func 𝐶) ≈ 1o → 𝐶 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat))
38	24, 25, 28, 37	istermo 18053	. . . . 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 18159	. . . . . . . 8 ⊢ ((((SetCat‘1o) Func 𝐶) ≈ 1o ∧ 𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)) → (𝑑(Hom ‘(CatCat‘{𝐶, (SetCat‘1o)}))𝐶) = (𝑑 Func 𝐶))
43	42	eleq2d 2855	. . . . . . 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 2620	. . . . . 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 3192	. . . . 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 50176	1 ⊢ (∀𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)∃!𝑓 𝑓 ∈ (𝑑 Func 𝐶) → 𝐶 ∈ TermCat)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ⊤wtru 1568  ∃wex 1806   ∈ wcel 2149  ∃!weu 2602   ≠ wne 2964  ∀wral 3085  Vcvv 3463   ∩ cin 3912  ∅c0 4294  {cpr 4596   class class class wbr 5113  ‘cfv 6537  (class class class)co 7411  1_oc1o 8445   ≈ cen 8939  Basecbs 17268  Hom chom 17320  Catccat 17719   Func cfunc 17910  TermOctermo 18038  SetCatcsetc 18131  CatCatccatc 18154  TermCatctermc 50134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-1st 7985  df-2nd 7986  df-supp 8156  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-map 8825  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-5 12305  df-6 12306  df-7 12307  df-8 12308  df-9 12309  df-n0 12504  df-z 12591  df-dec 12711  df-uz 12862  df-fz 13535  df-struct 17206  df-slot 17241  df-ndx 17253  df-base 17269  df-hom 17333  df-cco 17334  df-cat 17723  df-cid 17724  df-sect 17803  df-inv 17804  df-iso 17805  df-cic 17852  df-func 17914  df-idfu 17915  df-cofu 17916  df-full 17962  df-fth 17963  df-termo 18041  df-setc 18132  df-catc 18155  df-thinc 50080  df-termc 50135

This theorem is referenced by:  termc  50181