Theorem uobeqterm 49577

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 2731	. . . . 5 ⊢ (CatCat‘{𝐷, 𝐸}) = (CatCat‘{𝐷, 𝐸})
3		eqid 2731	. . . . 5 ⊢ (Base‘(CatCat‘{𝐷, 𝐸})) = (Base‘(CatCat‘{𝐷, 𝐸}))
4		uobeqterm.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ TermCat)
5		prid1g 4713	. . . . . . . 8 ⊢ (𝐷 ∈ TermCat → 𝐷 ∈ {𝐷, 𝐸})
6	4, 5	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ {𝐷, 𝐸})
7	4	termccd 49510	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
8	6, 7	elind 4150	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ({𝐷, 𝐸} ∩ Cat))
9		prex 5375	. . . . . . . 8 ⊢ {𝐷, 𝐸} ∈ V
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → {𝐷, 𝐸} ∈ V)
11	2, 3, 10	catcbas 18005	. . . . . 6 ⊢ (𝜑 → (Base‘(CatCat‘{𝐷, 𝐸})) = ({𝐷, 𝐸} ∩ Cat))
12	8, 11	eleqtrrd 2834	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (Base‘(CatCat‘{𝐷, 𝐸})))
13		prid2g 4714	. . . . . . . 8 ⊢ (𝐸 ∈ TermCat → 𝐸 ∈ {𝐷, 𝐸})
14	1, 13	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ {𝐷, 𝐸})
15	1	termccd 49510	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ Cat)
16	14, 15	elind 4150	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ({𝐷, 𝐸} ∩ Cat))
17	16, 11	eleqtrrd 2834	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (Base‘(CatCat‘{𝐷, 𝐸})))
18	2, 3, 12, 17, 4	termcciso 49547	. . . 4 ⊢ (𝜑 → (𝐸 ∈ TermCat ↔ 𝐷( ≃𝑐 ‘(CatCat‘{𝐷, 𝐸}))𝐸))
19	1, 18	mpbid 232	. . 3 ⊢ (𝜑 → 𝐷( ≃𝑐 ‘(CatCat‘{𝐷, 𝐸}))𝐸)
20		eqid 2731	. . . 4 ⊢ (Iso‘(CatCat‘{𝐷, 𝐸})) = (Iso‘(CatCat‘{𝐷, 𝐸}))
21	2	catccat 18012	. . . . 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 17703	. . 3 ⊢ (𝜑 → (𝐷( ≃𝑐 ‘(CatCat‘{𝐷, 𝐸}))𝐸 ↔ ∃𝑘 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)))
24	19, 23	mpbid 232	. 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 480	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → 𝑋 ∈ 𝐴)
28		uobeqterm.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
29	28	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → 𝐹 ∈ (𝐶 Func 𝐷))
30		fullfunc 17812	. . . . 5 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
31		uobeqterm.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐸)
32		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸))
33	2, 25, 31, 20, 32	catcisoi 49431	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → (𝑘 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ∧ (1st ‘𝑘):𝐴–1-1-onto→𝐵))
34	33	simpld 494	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → 𝑘 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
35	34	elin1d 4154	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → 𝑘 ∈ (𝐷 Full 𝐸))
36	30, 35	sselid 3932	. . . 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 480	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → 𝐺 ∈ (𝐶 Func 𝐸))
39	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → 𝐸 ∈ TermCat)
40	29, 36, 38, 39	cofuterm 49576	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → (𝑘 ∘func 𝐹) = 𝐺)
41	36	func1st2nd 49107	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → (1st ‘𝑘)(𝐷 Func 𝐸)(2nd ‘𝑘))
42	25, 31, 41	funcf1 17770	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → (1st ‘𝑘):𝐴⟶𝐵)
43	42, 27	ffvelcdmd 7018	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → ((1st ‘𝑘)‘𝑋) ∈ 𝐵)
44		uobeqterm.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
45	44	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → 𝑌 ∈ 𝐵)
46	39, 31, 43, 45	termcbasmo 49514	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → ((1st ‘𝑘)‘𝑋) = 𝑌)
47	25, 27, 29, 40, 46, 2, 20, 32	uobeq3 49433	. 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 1936	1 ⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  Vcvv 3436   ∩ cin 3901  {cpr 4578   class class class wbr 5091  dom cdm 5616  –1-1-onto→wf1o 6480  ‘cfv 6481  (class class class)co 7346  1^st c1st 7919  2^nd c2nd 7920  Basecbs 17117  Catccat 17567  Isociso 17650   ≃_𝑐 ccic 17699   Func cfunc 17758   Full cful 17808   Faith cfth 17809  CatCatccatc 18002   UP cup 49204  TermCatctermc 49503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-supp 8091  df-tpos 8156  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-map 8752  df-ixp 8822  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-nn 12123  df-2 12185  df-3 12186  df-4 12187  df-5 12188  df-6 12189  df-7 12190  df-8 12191  df-9 12192  df-n0 12379  df-z 12466  df-dec 12586  df-uz 12730  df-fz 13405  df-struct 17055  df-sets 17072  df-slot 17090  df-ndx 17102  df-base 17118  df-hom 17182  df-cco 17183  df-cat 17571  df-cid 17572  df-homf 17573  df-comf 17574  df-oppc 17615  df-sect 17651  df-inv 17652  df-iso 17653  df-cic 17700  df-func 17762  df-idfu 17763  df-cofu 17764  df-full 17810  df-fth 17811  df-nat 17850  df-fuc 17851  df-inito 17888  df-termo 17889  df-catc 18003  df-up 49205  df-thinc 49449  df-termc 49504

This theorem is referenced by:  isinito4  49578