uobeqterm - Mathbox for Zhi Wang

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Theorem uobeqterm 50043

Description: Universal objects and terminal categories. (Contributed by Zhi Wang, 17-Nov-2025.)

**Hypotheses**
Ref	Expression
uobeqterm.a	⊢ 𝐴 = (Base‘𝐷)
uobeqterm.b	⊢ 𝐵 = (Base‘𝐸)
uobeqterm.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
uobeqterm.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
uobeqterm.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
uobeqterm.g	⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))
uobeqterm.d	⊢ (𝜑 → 𝐷 ∈ TermCat)
uobeqterm.e	⊢ (𝜑 → 𝐸 ∈ TermCat)

**Assertion**
Ref	Expression
uobeqterm	⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))

Proof of Theorem uobeqterm Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uobeqterm.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ TermCat)
2		eqid 2740	. . . . 5 ⊢ (CatCat‘{𝐷, 𝐸}) = (CatCat‘{𝐷, 𝐸})
3		eqid 2740	. . . . 5 ⊢ (Base‘(CatCat‘{𝐷, 𝐸})) = (Base‘(CatCat‘{𝐷, 𝐸}))
4		uobeqterm.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ TermCat)
5		prid1g 4699	. . . . . . . 8 ⊢ (𝐷 ∈ TermCat → 𝐷 ∈ {𝐷, 𝐸})
6	4, 5	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ {𝐷, 𝐸})
7	4	termccd 49976	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
8	6, 7	elind 4136	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ({𝐷, 𝐸} ∩ Cat))
9		prex 5374	. . . . . . . 8 ⊢ {𝐷, 𝐸} ∈ V
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → {𝐷, 𝐸} ∈ V)
11	2, 3, 10	catcbas 18066	. . . . . 6 ⊢ (𝜑 → (Base‘(CatCat‘{𝐷, 𝐸})) = ({𝐷, 𝐸} ∩ Cat))
12	8, 11	eleqtrrd 2843	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (Base‘(CatCat‘{𝐷, 𝐸})))
13		prid2g 4700	. . . . . . . 8 ⊢ (𝐸 ∈ TermCat → 𝐸 ∈ {𝐷, 𝐸})
14	1, 13	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ {𝐷, 𝐸})
15	1	termccd 49976	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ Cat)
16	14, 15	elind 4136	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ({𝐷, 𝐸} ∩ Cat))
17	16, 11	eleqtrrd 2843	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (Base‘(CatCat‘{𝐷, 𝐸})))
18	2, 3, 12, 17, 4	termcciso 50013	. . . 4 ⊢ (𝜑 → (𝐸 ∈ TermCat ↔ 𝐷( ≃_𝑐 ‘(CatCat‘{𝐷, 𝐸}))𝐸))
19	1, 18	mpbid 233	. . 3 ⊢ (𝜑 → 𝐷( ≃_𝑐 ‘(CatCat‘{𝐷, 𝐸}))𝐸)
20		eqid 2740	. . . 4 ⊢ (Iso‘(CatCat‘{𝐷, 𝐸})) = (Iso‘(CatCat‘{𝐷, 𝐸}))
21	2	catccat 18073	. . . . 5 ⊢ ({𝐷, 𝐸} ∈ V → (CatCat‘{𝐷, 𝐸}) ∈ Cat)
22	10, 21	syl 17	. . . 4 ⊢ (𝜑 → (CatCat‘{𝐷, 𝐸}) ∈ Cat)
23	20, 3, 22, 12, 17	cic 17764	. . 3 ⊢ (𝜑 → (𝐷( ≃_𝑐 ‘(CatCat‘{𝐷, 𝐸}))𝐸 ↔ ∃𝑘 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)))
24	19, 23	mpbid 233	. 2 ⊢ (𝜑 → ∃𝑘 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸))
25		uobeqterm.a	. . 3 ⊢ 𝐴 = (Base‘𝐷)
26		uobeqterm.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
27	26	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → 𝑋 ∈ 𝐴)
28		uobeqterm.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
29	28	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → 𝐹 ∈ (𝐶 Func 𝐷))
30		fullfunc 17873	. . . . 5 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
31		uobeqterm.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐸)
32		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸))
33	2, 25, 31, 20, 32	catcisoi 49897	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → (𝑘 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ∧ (1^st ‘𝑘):𝐴–1-1-onto→𝐵))
34	33	simpld 495	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → 𝑘 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
35	34	elin1d 4140	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → 𝑘 ∈ (𝐷 Full 𝐸))
36	30, 35	sselid 3920	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → 𝑘 ∈ (𝐷 Func 𝐸))
37		uobeqterm.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))
38	37	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → 𝐺 ∈ (𝐶 Func 𝐸))
39	1	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → 𝐸 ∈ TermCat)
40	29, 36, 38, 39	cofuterm 50042	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → (𝑘 ∘_func 𝐹) = 𝐺)
41	36	func1st2nd 49573	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → (1^st ‘𝑘)(𝐷 Func 𝐸)(2^nd ‘𝑘))
42	25, 31, 41	funcf1 17831	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → (1^st ‘𝑘):𝐴⟶𝐵)
43	42, 27	ffvelcdmd 7033	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → ((1^st ‘𝑘)‘𝑋) ∈ 𝐵)
44		uobeqterm.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
45	44	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → 𝑌 ∈ 𝐵)
46	39, 31, 43, 45	termcbasmo 49980	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → ((1^st ‘𝑘)‘𝑋) = 𝑌)
47	25, 27, 29, 40, 46, 2, 20, 32	uobeq3 49899	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))
48	24, 47	exlimddv 1942	1 ⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∃wex 1786 ∈ wcel 2119 Vcvv 3432 ∩ cin 3889 {cpr 4564 class class class wbr 5079 dom cdm 5625 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7363 1^st c1st 7936 2^nd c2nd 7937 Basecbs 17177 Catccat 17628 Isociso 17711 ≃_𝑐 ccic 17760 Func cfunc 17819 Full cful 17869 Faith cfth 17870 CatCatccatc 18063 UP cup 49670 TermCatctermc 49969

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 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 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-tpos 8173 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-er 8640 df-map 8772 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-fz 13460 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-hom 17242 df-cco 17243 df-cat 17632 df-cid 17633 df-homf 17634 df-comf 17635 df-oppc 17676 df-sect 17712 df-inv 17713 df-iso 17714 df-cic 17761 df-func 17823 df-idfu 17824 df-cofu 17825 df-full 17871 df-fth 17872 df-nat 17911 df-fuc 17912 df-inito 17949 df-termo 17950 df-catc 18064 df-up 49671 df-thinc 49915 df-termc 49970

This theorem is referenced by: isinito4 50044

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