Theorem termc2 49487

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 49488 for hints. See also eufunc 49491 and euendfunc2 49496 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 2730	. 2 ⊢ (CatCat‘{𝐶, (SetCat‘1o)}) = (CatCat‘{𝐶, (SetCat‘1o)})
2		fvex 6873	. . . . . 6 ⊢ (SetCat‘1o) ∈ V
3	2	prid2 4729	. . . . 5 ⊢ (SetCat‘1o) ∈ {𝐶, (SetCat‘1o)}
4		setc1oterm 49460	. . . . 5 ⊢ (SetCat‘1o) ∈ TermCat
5	3, 4	elini 4164	. . . 4 ⊢ (SetCat‘1o) ∈ ({𝐶, (SetCat‘1o)} ∩ TermCat)
6	5	ne0ii 4309	. . 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 49448	. . . . . . . 8 ⊢ (⊤ → (SetCat‘1o) ∈ Cat)
10	9	mptru 1547	. . . . . . 7 ⊢ (SetCat‘1o) ∈ Cat
11	3, 10	elini 4164	. . . . . 6 ⊢ (SetCat‘1o) ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)
12		oveq1 7396	. . . . . . . . 9 ⊢ (𝑑 = (SetCat‘1o) → (𝑑 Func 𝐶) = ((SetCat‘1o) Func 𝐶))
13	12	eleq2d 2815	. . . . . . . 8 ⊢ (𝑑 = (SetCat‘1o) → (𝑓 ∈ (𝑑 Func 𝐶) ↔ 𝑓 ∈ ((SetCat‘1o) Func 𝐶)))
14	13	eubidv 2580	. . . . . . 7 ⊢ (𝑑 = (SetCat‘1o) → (∃!𝑓 𝑓 ∈ (𝑑 Func 𝐶) ↔ ∃!𝑓 𝑓 ∈ ((SetCat‘1o) Func 𝐶)))
15	14	rspcv 3587	. . . . . 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 9001	. . . . 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 2730	. . . . . . . . 9 ⊢ (Base‘(CatCat‘{𝐶, (SetCat‘1o)})) = (Base‘(CatCat‘{𝐶, (SetCat‘1o)}))
20		prex 5394	. . . . . . . . . 10 ⊢ {𝐶, (SetCat‘1o)} ∈ V
21	20	a1i 11	. . . . . . . . 9 ⊢ (⊤ → {𝐶, (SetCat‘1o)} ∈ V)
22	1, 19, 21	catcbas 18069	. . . . . . . 8 ⊢ (⊤ → (Base‘(CatCat‘{𝐶, (SetCat‘1o)})) = ({𝐶, (SetCat‘1o)} ∩ Cat))
23	22	mptru 1547	. . . . . . 7 ⊢ (Base‘(CatCat‘{𝐶, (SetCat‘1o)})) = ({𝐶, (SetCat‘1o)} ∩ Cat)
24	23	eqcomi 2739	. . . . . 6 ⊢ ({𝐶, (SetCat‘1o)} ∩ Cat) = (Base‘(CatCat‘{𝐶, (SetCat‘1o)}))
25		eqid 2730	. . . . . 6 ⊢ (Hom ‘(CatCat‘{𝐶, (SetCat‘1o)})) = (Hom ‘(CatCat‘{𝐶, (SetCat‘1o)}))
26	1	catccat 18076	. . . . . . . 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 2571	. . . . . . . . . 10 ⊢ (∃!𝑓 𝑓 ∈ ((SetCat‘1o) Func 𝐶) → ∃𝑓 𝑓 ∈ ((SetCat‘1o) Func 𝐶))
30		funcrcl 17831	. . . . . . . . . . . 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 1930	. . . . . . . . . 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 4726	. . . . . . . 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 4165	. . . . . 6 ⊢ (((SetCat‘1o) Func 𝐶) ≈ 1o → 𝐶 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat))
38	24, 25, 28, 37	istermo 17965	. . . . 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 18071	. . . . . . . 8 ⊢ ((((SetCat‘1o) Func 𝐶) ≈ 1o ∧ 𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)) → (𝑑(Hom ‘(CatCat‘{𝐶, (SetCat‘1o)}))𝐶) = (𝑑 Func 𝐶))
43	42	eleq2d 2815	. . . . . . 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 2580	. . . . . 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 3155	. . . . 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 49483	1 ⊢ (∀𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)∃!𝑓 𝑓 ∈ (𝑑 Func 𝐶) → 𝐶 ∈ TermCat)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ⊤wtru 1541  ∃wex 1779   ∈ wcel 2109  ∃!weu 2562   ≠ wne 2926  ∀wral 3045  Vcvv 3450   ∩ cin 3915  ∅c0 4298  {cpr 4593   class class class wbr 5109  ‘cfv 6513  (class class class)co 7389  1_oc1o 8429   ≈ cen 8917  Basecbs 17185  Hom chom 17237  Catccat 17631   Func cfunc 17822  TermOctermo 17950  SetCatcsetc 18043  CatCatccatc 18066  TermCatctermc 49441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-om 7845  df-1st 7970  df-2nd 7971  df-supp 8142  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-1o 8436  df-er 8673  df-map 8803  df-ixp 8873  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-pnf 11216  df-mnf 11217  df-xr 11218  df-ltxr 11219  df-le 11220  df-sub 11413  df-neg 11414  df-nn 12188  df-2 12250  df-3 12251  df-4 12252  df-5 12253  df-6 12254  df-7 12255  df-8 12256  df-9 12257  df-n0 12449  df-z 12536  df-dec 12656  df-uz 12800  df-fz 13475  df-struct 17123  df-slot 17158  df-ndx 17170  df-base 17186  df-hom 17250  df-cco 17251  df-cat 17635  df-cid 17636  df-sect 17715  df-inv 17716  df-iso 17717  df-cic 17764  df-func 17826  df-idfu 17827  df-cofu 17828  df-full 17874  df-fth 17875  df-termo 17953  df-setc 18044  df-catc 18067  df-thinc 49387  df-termc 49442

This theorem is referenced by:  termc  49488