Theorem uobeqterm 49899

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 2737	. . . . 5 ⊢ (CatCat‘{𝐷, 𝐸}) = (CatCat‘{𝐷, 𝐸})
3		eqid 2737	. . . . 5 ⊢ (Base‘(CatCat‘{𝐷, 𝐸})) = (Base‘(CatCat‘{𝐷, 𝐸}))
4		uobeqterm.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ TermCat)
5		prid1g 4719	. . . . . . . 8 ⊢ (𝐷 ∈ TermCat → 𝐷 ∈ {𝐷, 𝐸})
6	4, 5	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ {𝐷, 𝐸})
7	4	termccd 49832	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
8	6, 7	elind 4154	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ({𝐷, 𝐸} ∩ Cat))
9		prex 5384	. . . . . . . 8 ⊢ {𝐷, 𝐸} ∈ V
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → {𝐷, 𝐸} ∈ V)
11	2, 3, 10	catcbas 18037	. . . . . 6 ⊢ (𝜑 → (Base‘(CatCat‘{𝐷, 𝐸})) = ({𝐷, 𝐸} ∩ Cat))
12	8, 11	eleqtrrd 2840	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (Base‘(CatCat‘{𝐷, 𝐸})))
13		prid2g 4720	. . . . . . . 8 ⊢ (𝐸 ∈ TermCat → 𝐸 ∈ {𝐷, 𝐸})
14	1, 13	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ {𝐷, 𝐸})
15	1	termccd 49832	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ Cat)
16	14, 15	elind 4154	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ({𝐷, 𝐸} ∩ Cat))
17	16, 11	eleqtrrd 2840	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (Base‘(CatCat‘{𝐷, 𝐸})))
18	2, 3, 12, 17, 4	termcciso 49869	. . . 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 2737	. . . 4 ⊢ (Iso‘(CatCat‘{𝐷, 𝐸})) = (Iso‘(CatCat‘{𝐷, 𝐸}))
21	2	catccat 18044	. . . . 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 17735	. . 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 17844	. . . . 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 49753	. . . . . . 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 4158	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → 𝑘 ∈ (𝐷 Full 𝐸))
36	30, 35	sselid 3933	. . . 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 49898	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → (𝑘 ∘func 𝐹) = 𝐺)
41	36	func1st2nd 49429	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → (1st ‘𝑘)(𝐷 Func 𝐸)(2nd ‘𝑘))
42	25, 31, 41	funcf1 17802	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → (1st ‘𝑘):𝐴⟶𝐵)
43	42, 27	ffvelcdmd 7039	. . . 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 49836	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷(Iso‘(CatCat‘{𝐷, 𝐸}))𝐸)) → ((1st ‘𝑘)‘𝑋) = 𝑌)
47	25, 27, 29, 40, 46, 2, 20, 32	uobeq3 49755	. 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 1937	1 ⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3442   ∩ cin 3902  {cpr 4584   class class class wbr 5100  dom cdm 5632  –1-1-onto→wf1o 6499  ‘cfv 6500  (class class class)co 7368  1^st c1st 7941  2^nd c2nd 7942  Basecbs 17148  Catccat 17599  Isociso 17682   ≃_𝑐 ccic 17731   Func cfunc 17790   Full cful 17840   Faith cfth 17841  CatCatccatc 18034   UP cup 49526  TermCatctermc 49825

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-tpos 8178  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-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-hom 17213  df-cco 17214  df-cat 17603  df-cid 17604  df-homf 17605  df-comf 17606  df-oppc 17647  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-nat 17882  df-fuc 17883  df-inito 17920  df-termo 17921  df-catc 18035  df-up 49527  df-thinc 49771  df-termc 49826

This theorem is referenced by:  isinito4  49900