termcterm2 - Mathbox for Zhi Wang

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Theorem termcterm2 49212

Description: A terminal object of the category of small categories is a terminal category. (Contributed by Zhi Wang, 18-Oct-2025.) (Proof shortened by Zhi Wang, 23-Oct-2025.)

**Hypotheses**
Ref	Expression
termcterm.e	⊢ 𝐸 = (CatCat‘𝑈)
termcterm2.	⊢ (𝜑 → (𝑈 ∩ TermCat) ≠ ∅)
termcterm2.t	⊢ (𝜑 → 𝐶 ∈ (TermO‘𝐸))

**Assertion**
Ref	Expression
termcterm2	⊢ (𝜑 → 𝐶 ∈ TermCat)

Proof of Theorem termcterm2 Dummy variables 𝑑 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		termcterm2.	. . 3 ⊢ (𝜑 → (𝑈 ∩ TermCat) ≠ ∅)
2		n0 4333	. . 3 ⊢ ((𝑈 ∩ TermCat) ≠ ∅ ↔ ∃𝑑 𝑑 ∈ (𝑈 ∩ TermCat))
3	1, 2	sylib 218	. 2 ⊢ (𝜑 → ∃𝑑 𝑑 ∈ (𝑈 ∩ TermCat))
4		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝑑 ∈ (𝑈 ∩ TermCat))
5	4	elin2d 4185	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝑑 ∈ TermCat)
6	5	termcthind 49177	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝑑 ∈ ThinCat)
7		termcterm.e	. . . . 5 ⊢ 𝐸 = (CatCat‘𝑈)
8		eqid 2734	. . . . 5 ⊢ (Base‘𝐸) = (Base‘𝐸)
9		termcterm2.t	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ (TermO‘𝐸))
10	9	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝐶 ∈ (TermO‘𝐸))
11	8	termoo2 48984	. . . . . . 7 ⊢ (𝐶 ∈ (TermO‘𝐸) → 𝐶 ∈ (Base‘𝐸))
12	10, 11	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝐶 ∈ (Base‘𝐸))
13	7, 8	elbasfv 17236	. . . . . 6 ⊢ (𝐶 ∈ (Base‘𝐸) → 𝑈 ∈ V)
14	12, 13	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝑈 ∈ V)
15	4	elin1d 4184	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝑑 ∈ 𝑈)
16	5	termccd 49178	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝑑 ∈ Cat)
17	15, 16	elind 4180	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝑑 ∈ (𝑈 ∩ Cat))
18	7, 8, 14	catcbas 18118	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → (Base‘𝐸) = (𝑈 ∩ Cat))
19	17, 18	eleqtrrd 2836	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝑑 ∈ (Base‘𝐸))
20		termorcl 18008	. . . . . . 7 ⊢ (𝐶 ∈ (TermO‘𝐸) → 𝐸 ∈ Cat)
21	10, 20	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝐸 ∈ Cat)
22	7, 14, 15, 5	termcterm 49211	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝑑 ∈ (TermO‘𝐸))
23	21, 10, 22	termoeu1w 18036	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝐶( ≃_𝑐 ‘𝐸)𝑑)
24	7, 8, 14, 12, 19, 23	thincciso4 49158	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → (𝐶 ∈ ThinCat ↔ 𝑑 ∈ ThinCat))
25	6, 24	mpbird 257	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝐶 ∈ ThinCat)
26	21, 10, 22	termoeu1 18035	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → ∃!𝑓 𝑓 ∈ (𝐶(Iso‘𝐸)𝑑))
27		euex 2575	. . . . . 6 ⊢ (∃!𝑓 𝑓 ∈ (𝐶(Iso‘𝐸)𝑑) → ∃𝑓 𝑓 ∈ (𝐶(Iso‘𝐸)𝑑))
28	26, 27	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → ∃𝑓 𝑓 ∈ (𝐶(Iso‘𝐸)𝑑))
29		eqid 2734	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
30		eqid 2734	. . . . . . . 8 ⊢ (Base‘𝑑) = (Base‘𝑑)
31		eqid 2734	. . . . . . . 8 ⊢ (Iso‘𝐸) = (Iso‘𝐸)
32	7, 8, 29, 30, 14, 12, 19, 31	catciso 18128	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → (𝑓 ∈ (𝐶(Iso‘𝐸)𝑑) ↔ (𝑓 ∈ ((𝐶 Full 𝑑) ∩ (𝐶 Faith 𝑑)) ∧ (1^st ‘𝑓):(Base‘𝐶)–1-1-onto→(Base‘𝑑))))
33	32	simplbda 499	. . . . . 6 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) ∧ 𝑓 ∈ (𝐶(Iso‘𝐸)𝑑)) → (1^st ‘𝑓):(Base‘𝐶)–1-1-onto→(Base‘𝑑))
34		fvex 6899	. . . . . . 7 ⊢ (Base‘𝐶) ∈ V
35	34	f1oen 8995	. . . . . 6 ⊢ ((1^st ‘𝑓):(Base‘𝐶)–1-1-onto→(Base‘𝑑) → (Base‘𝐶) ≈ (Base‘𝑑))
36	33, 35	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) ∧ 𝑓 ∈ (𝐶(Iso‘𝐸)𝑑)) → (Base‘𝐶) ≈ (Base‘𝑑))
37	28, 36	exlimddv 1934	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → (Base‘𝐶) ≈ (Base‘𝑑))
38	30	istermc3 49175	. . . . . 6 ⊢ (𝑑 ∈ TermCat ↔ (𝑑 ∈ ThinCat ∧ (Base‘𝑑) ≈ 1_o))
39	5, 38	sylib 218	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → (𝑑 ∈ ThinCat ∧ (Base‘𝑑) ≈ 1_o))
40	39	simprd 495	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → (Base‘𝑑) ≈ 1_o)
41		entr 9028	. . . 4 ⊢ (((Base‘𝐶) ≈ (Base‘𝑑) ∧ (Base‘𝑑) ≈ 1_o) → (Base‘𝐶) ≈ 1_o)
42	37, 40, 41	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → (Base‘𝐶) ≈ 1_o)
43	29	istermc3 49175	. . 3 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ (Base‘𝐶) ≈ 1_o))
44	25, 42, 43	sylanbrc 583	. 2 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝐶 ∈ TermCat)
45	3, 44	exlimddv 1934	1 ⊢ (𝜑 → 𝐶 ∈ TermCat)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ∃!weu 2566 ≠ wne 2931 Vcvv 3463 ∩ cin 3930 ∅c0 4313 class class class wbr 5123 –1-1-onto→wf1o 6540 ‘cfv 6541 (class class class)co 7413 1^st c1st 7994 1_oc1o 8481 ≈ cen 8964 Basecbs 17230 Catccat 17679 Isociso 17762 Full cful 17921 Faith cfth 17922 TermOctermo 17999 CatCatccatc 18115 ThinCatcthinc 49118 TermCatctermc 49171

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 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 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-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4888 df-iun 4973 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7870 df-1st 7996 df-2nd 7997 df-supp 8168 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8727 df-map 8850 df-ixp 8920 df-en 8968 df-dom 8969 df-sdom 8970 df-fin 8971 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12510 df-z 12597 df-dec 12717 df-uz 12861 df-fz 13530 df-struct 17167 df-slot 17202 df-ndx 17214 df-base 17231 df-hom 17298 df-cco 17299 df-cat 17683 df-cid 17684 df-sect 17763 df-inv 17764 df-iso 17765 df-cic 17812 df-func 17875 df-idfu 17876 df-cofu 17877 df-full 17923 df-fth 17924 df-termo 18002 df-catc 18116 df-thinc 49119 df-termc 49172

This theorem is referenced by: termcterm3 49213 termcciso 49214 termc2 49216

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