termcterm2 - Mathbox for Zhi Wang

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Theorem termcterm2 49873

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 4307	. . 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 4159	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝑑 ∈ TermCat)
6	5	termcthind 49837	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝑑 ∈ ThinCat)
7		termcterm.e	. . . . 5 ⊢ 𝐸 = (CatCat‘𝑈)
8		eqid 2737	. . . . 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 49592	. . . . . . 7 ⊢ (𝐶 ∈ (TermO‘𝐸) → 𝐶 ∈ (Base‘𝐸))
12	10, 11	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝐶 ∈ (Base‘𝐸))
13	7, 8	elbasfv 17154	. . . . . 6 ⊢ (𝐶 ∈ (Base‘𝐸) → 𝑈 ∈ V)
14	12, 13	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝑈 ∈ V)
15	4	elin1d 4158	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝑑 ∈ 𝑈)
16	5	termccd 49838	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝑑 ∈ Cat)
17	15, 16	elind 4154	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝑑 ∈ (𝑈 ∩ Cat))
18	7, 8, 14	catcbas 18037	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → (Base‘𝐸) = (𝑈 ∩ Cat))
19	17, 18	eleqtrrd 2840	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝑑 ∈ (Base‘𝐸))
20		termorcl 17927	. . . . . . 7 ⊢ (𝐶 ∈ (TermO‘𝐸) → 𝐸 ∈ Cat)
21	10, 20	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝐸 ∈ Cat)
22	7, 14, 15, 5	termcterm 49872	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝑑 ∈ (TermO‘𝐸))
23	21, 10, 22	termoeu1w 17955	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝐶( ≃_𝑐 ‘𝐸)𝑑)
24	7, 8, 14, 12, 19, 23	thincciso4 49816	. . . 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 17954	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → ∃!𝑓 𝑓 ∈ (𝐶(Iso‘𝐸)𝑑))
27		euex 2578	. . . . . 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 2737	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
30		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝑑) = (Base‘𝑑)
31		eqid 2737	. . . . . . . 8 ⊢ (Iso‘𝐸) = (Iso‘𝐸)
32	7, 8, 29, 30, 14, 12, 19, 31	catciso 18047	. . . . . . 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 6855	. . . . . . 7 ⊢ (Base‘𝐶) ∈ V
35	34	f1oen 8921	. . . . . 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 1937	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → (Base‘𝐶) ≈ (Base‘𝑑))
38	30	istermc3 49835	. . . . . 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 8955	. . . 4 ⊢ (((Base‘𝐶) ≈ (Base‘𝑑) ∧ (Base‘𝑑) ≈ 1_o) → (Base‘𝐶) ≈ 1_o)
42	37, 40, 41	syl2anc 585	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → (Base‘𝐶) ≈ 1_o)
43	29	istermc3 49835	. . 3 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ (Base‘𝐶) ≈ 1_o))
44	25, 42, 43	sylanbrc 584	. 2 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝑈 ∩ TermCat)) → 𝐶 ∈ TermCat)
45	3, 44	exlimddv 1937	1 ⊢ (𝜑 → 𝐶 ∈ TermCat)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∃wex 1781 ∈ wcel 2114 ∃!weu 2569 ≠ wne 2933 Vcvv 3442 ∩ cin 3902 ∅c0 4287 class class class wbr 5100 –1-1-onto→wf1o 6499 ‘cfv 6500 (class class class)co 7368 1^st c1st 7941 1_oc1o 8400 ≈ cen 8892 Basecbs 17148 Catccat 17599 Isociso 17682 Full cful 17840 Faith cfth 17841 TermOctermo 17918 CatCatccatc 18034 ThinCatcthinc 49776 TermCatctermc 49831

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 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

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 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-fz 13436 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17149 df-hom 17213 df-cco 17214 df-cat 17603 df-cid 17604 df-sect 17683 df-inv 17684 df-iso 17685 df-cic 17732 df-func 17794 df-idfu 17795 df-cofu 17796 df-full 17842 df-fth 17843 df-termo 17921 df-catc 18035 df-thinc 49777 df-termc 49832

This theorem is referenced by: termcterm3 49874 termcciso 49875 termc2 49877

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