Theorem termc2 50005

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 50006 for hints. See also eufunc 50009 and euendfunc2 50014 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 2737	. 2 ⊢ (CatCat‘{𝐶, (SetCat‘1o)}) = (CatCat‘{𝐶, (SetCat‘1o)})
2		fvex 6847	. . . . . 6 ⊢ (SetCat‘1o) ∈ V
3	2	prid2 4708	. . . . 5 ⊢ (SetCat‘1o) ∈ {𝐶, (SetCat‘1o)}
4		setc1oterm 49978	. . . . 5 ⊢ (SetCat‘1o) ∈ TermCat
5	3, 4	elini 4140	. . . 4 ⊢ (SetCat‘1o) ∈ ({𝐶, (SetCat‘1o)} ∩ TermCat)
6	5	ne0ii 4285	. . 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 49966	. . . . . . . 8 ⊢ (⊤ → (SetCat‘1o) ∈ Cat)
10	9	mptru 1549	. . . . . . 7 ⊢ (SetCat‘1o) ∈ Cat
11	3, 10	elini 4140	. . . . . 6 ⊢ (SetCat‘1o) ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)
12		oveq1 7367	. . . . . . . . 9 ⊢ (𝑑 = (SetCat‘1o) → (𝑑 Func 𝐶) = ((SetCat‘1o) Func 𝐶))
13	12	eleq2d 2823	. . . . . . . 8 ⊢ (𝑑 = (SetCat‘1o) → (𝑓 ∈ (𝑑 Func 𝐶) ↔ 𝑓 ∈ ((SetCat‘1o) Func 𝐶)))
14	13	eubidv 2587	. . . . . . 7 ⊢ (𝑑 = (SetCat‘1o) → (∃!𝑓 𝑓 ∈ (𝑑 Func 𝐶) ↔ ∃!𝑓 𝑓 ∈ ((SetCat‘1o) Func 𝐶)))
15	14	rspcv 3561	. . . . . 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 8968	. . . . 5 ⊢ (((SetCat‘1o) Func 𝐶) ≈ 1o ↔ ∃!𝑓 𝑓 ∈ ((SetCat‘1o) Func 𝐶))
18	16, 17	sylibr 234	. . . 4 ⊢ (∀𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)∃!𝑓 𝑓 ∈ (𝑑 Func 𝐶) → ((SetCat‘1o) Func 𝐶) ≈ 1o)
19		eqid 2737	. . . . . . . . 9 ⊢ (Base‘(CatCat‘{𝐶, (SetCat‘1o)})) = (Base‘(CatCat‘{𝐶, (SetCat‘1o)}))
20		prex 5375	. . . . . . . . . 10 ⊢ {𝐶, (SetCat‘1o)} ∈ V
21	20	a1i 11	. . . . . . . . 9 ⊢ (⊤ → {𝐶, (SetCat‘1o)} ∈ V)
22	1, 19, 21	catcbas 18059	. . . . . . . 8 ⊢ (⊤ → (Base‘(CatCat‘{𝐶, (SetCat‘1o)})) = ({𝐶, (SetCat‘1o)} ∩ Cat))
23	22	mptru 1549	. . . . . . 7 ⊢ (Base‘(CatCat‘{𝐶, (SetCat‘1o)})) = ({𝐶, (SetCat‘1o)} ∩ Cat)
24	23	eqcomi 2746	. . . . . 6 ⊢ ({𝐶, (SetCat‘1o)} ∩ Cat) = (Base‘(CatCat‘{𝐶, (SetCat‘1o)}))
25		eqid 2737	. . . . . 6 ⊢ (Hom ‘(CatCat‘{𝐶, (SetCat‘1o)})) = (Hom ‘(CatCat‘{𝐶, (SetCat‘1o)}))
26	1	catccat 18066	. . . . . . . 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 2578	. . . . . . . . . 10 ⊢ (∃!𝑓 𝑓 ∈ ((SetCat‘1o) Func 𝐶) → ∃𝑓 𝑓 ∈ ((SetCat‘1o) Func 𝐶))
30		funcrcl 17821	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ ((SetCat‘1o) Func 𝐶) → ((SetCat‘1o) ∈ Cat ∧ 𝐶 ∈ Cat))
31	30	simprd 495	. . . . . . . . . . 11 ⊢ (𝑓 ∈ ((SetCat‘1o) Func 𝐶) → 𝐶 ∈ Cat)
32	31	exlimiv 1932	. . . . . . . . . 10 ⊢ (∃𝑓 𝑓 ∈ ((SetCat‘1o) Func 𝐶) → 𝐶 ∈ Cat)
33	29, 32	syl 17	. . . . . . . . 9 ⊢ (∃!𝑓 𝑓 ∈ ((SetCat‘1o) Func 𝐶) → 𝐶 ∈ Cat)
34	17, 33	sylbi 217	. . . . . . . 8 ⊢ (((SetCat‘1o) Func 𝐶) ≈ 1o → 𝐶 ∈ Cat)
35		prid1g 4705	. . . . . . . 8 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ {𝐶, (SetCat‘1o)})
36	34, 35	syl 17	. . . . . . 7 ⊢ (((SetCat‘1o) Func 𝐶) ≈ 1o → 𝐶 ∈ {𝐶, (SetCat‘1o)})
37	36, 34	elind 4141	. . . . . 6 ⊢ (((SetCat‘1o) Func 𝐶) ≈ 1o → 𝐶 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat))
38	24, 25, 28, 37	istermo 17955	. . . . 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 484	. . . . . . . . 9 ⊢ ((((SetCat‘1o) Func 𝐶) ≈ 1o ∧ 𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)) → 𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat))
41	37	adantr 480	. . . . . . . . 9 ⊢ ((((SetCat‘1o) Func 𝐶) ≈ 1o ∧ 𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)) → 𝐶 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat))
42	1, 24, 39, 25, 40, 41	catchom 18061	. . . . . . . 8 ⊢ ((((SetCat‘1o) Func 𝐶) ≈ 1o ∧ 𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)) → (𝑑(Hom ‘(CatCat‘{𝐶, (SetCat‘1o)}))𝐶) = (𝑑 Func 𝐶))
43	42	eleq2d 2823	. . . . . . 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 2587	. . . . . 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 3159	. . . . 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 279	. . . 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 17	. . 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 268	. 2 ⊢ (∀𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)∃!𝑓 𝑓 ∈ (𝑑 Func 𝐶) → 𝐶 ∈ (TermO‘(CatCat‘{𝐶, (SetCat‘1o)})))
49	1, 7, 48	termcterm2 50001	1 ⊢ (∀𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)∃!𝑓 𝑓 ∈ (𝑑 Func 𝐶) → 𝐶 ∈ TermCat)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ⊤wtru 1543  ∃wex 1781   ∈ wcel 2114  ∃!weu 2569   ≠ wne 2933  ∀wral 3052  Vcvv 3430   ∩ cin 3889  ∅c0 4274  {cpr 4570   class class class wbr 5086  ‘cfv 6492  (class class class)co 7360  1_oc1o 8391   ≈ cen 8883  Basecbs 17170  Hom chom 17222  Catccat 17621   Func cfunc 17812  TermOctermo 17940  SetCatcsetc 18033  CatCatccatc 18056  TermCatctermc 49959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8104  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-er 8636  df-map 8768  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-struct 17108  df-slot 17143  df-ndx 17155  df-base 17171  df-hom 17235  df-cco 17236  df-cat 17625  df-cid 17626  df-sect 17705  df-inv 17706  df-iso 17707  df-cic 17754  df-func 17816  df-idfu 17817  df-cofu 17818  df-full 17864  df-fth 17865  df-termo 17943  df-setc 18034  df-catc 18057  df-thinc 49905  df-termc 49960

This theorem is referenced by:  termc  50006