Theorem uobeqw 49125

Description: If a full functor (in fact, a full embedding) is a section of a fully faithful functor (surjective on objects), then the sets of universal objects are equal. (Contributed by Zhi Wang, 17-Nov-2025.)

Hypotheses

Ref	Expression
uobeq.b	⊢ 𝐵 = (Base‘𝐷)
uobeq.i	⊢ 𝐼 = (idfunc‘𝐷)
uobeq.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
uobeq.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
uobeq.k	⊢ (𝜑 → 𝐾 ∈ (𝐷 Full 𝐸))
uobeq.g	⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)
uobeq.y	⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)
uobeq.n	⊢ (𝜑 → (𝐿 ∘func 𝐾) = 𝐼)
uobeqw.l	⊢ (𝜑 → 𝐿 ∈ ((𝐸 Full 𝐷) ∩ (𝐸 Faith 𝐷)))

Assertion

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

Proof of Theorem uobeqw

Dummy variables 𝑚 𝑛 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.42v 1953	. . . . 5 ⊢ (∃𝑚(𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) ↔ (𝜑 ∧ ∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚))
2		fvexd 6880	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑧))‘𝑚) ∈ V)
3		uobeq.y	. . . . . . . . 9 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)
4	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → ((1st ‘𝐾)‘𝑋) = 𝑌)
5		uobeq.k	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Full 𝐸))
6		relfunc 17830	. . . . . . . . . . . 12 ⊢ Rel (𝐷 Func 𝐸)
7		fullfunc 17876	. . . . . . . . . . . . 13 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
8	7, 5	sselid 3952	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
9		1st2nd 8027	. . . . . . . . . . . 12 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)) → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
10	6, 8, 9	sylancr 587	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
11		uobeq.i	. . . . . . . . . . . . 13 ⊢ 𝐼 = (idfunc‘𝐷)
12	8	func1st2nd 48993	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾))
13		inss1 4208	. . . . . . . . . . . . . . . 16 ⊢ ((𝐸 Full 𝐷) ∩ (𝐸 Faith 𝐷)) ⊆ (𝐸 Full 𝐷)
14		fullfunc 17876	. . . . . . . . . . . . . . . 16 ⊢ (𝐸 Full 𝐷) ⊆ (𝐸 Func 𝐷)
15	13, 14	sstri 3964	. . . . . . . . . . . . . . 15 ⊢ ((𝐸 Full 𝐷) ∩ (𝐸 Faith 𝐷)) ⊆ (𝐸 Func 𝐷)
16		uobeqw.l	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐿 ∈ ((𝐸 Full 𝐷) ∩ (𝐸 Faith 𝐷)))
17	15, 16	sselid 3952	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐿 ∈ (𝐸 Func 𝐷))
18	17	func1st2nd 48993	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1st ‘𝐿)(𝐸 Func 𝐷)(2nd ‘𝐿))
19	8, 17	cofu1st2nd 49009	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐿 ∘func 𝐾) = (⟨(1st ‘𝐿), (2nd ‘𝐿)⟩ ∘func ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩))
20		uobeq.n	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐿 ∘func 𝐾) = 𝐼)
21	19, 20	eqtr3d 2767	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⟨(1st ‘𝐿), (2nd ‘𝐿)⟩ ∘func ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩) = 𝐼)
22	11, 12, 18, 21	cofidfth 49073	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Faith 𝐸)(2nd ‘𝐾))
23		df-br 5116	. . . . . . . . . . . 12 ⊢ ((1st ‘𝐾)(𝐷 Faith 𝐸)(2nd ‘𝐾) ↔ ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩ ∈ (𝐷 Faith 𝐸))
24	22, 23	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩ ∈ (𝐷 Faith 𝐸))
25	10, 24	eqeltrd 2829	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Faith 𝐸))
26	5, 25	elind 4171	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
27	26	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
28		uobeq.g	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)
29	28	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → (𝐾 ∘func 𝐹) = 𝐺)
30		eqidd 2731	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑧))‘𝑚) = ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑧))‘𝑚))
31		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚)
32	4, 27, 29, 30, 31	uptrai 49124	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → 𝑧(𝐺(𝐶 UP 𝐸)𝑌)((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑧))‘𝑚))
33		breq2 5119	. . . . . . 7 ⊢ (𝑛 = ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑧))‘𝑚) → (𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛 ↔ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑧))‘𝑚)))
34	2, 32, 33	spcedv 3573	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛)
35	34	exlimiv 1930	. . . . 5 ⊢ (∃𝑚(𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛)
36	1, 35	sylbir 235	. . . 4 ⊢ ((𝜑 ∧ ∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛)
37		19.42v 1953	. . . . 5 ⊢ (∃𝑛(𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) ↔ (𝜑 ∧ ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛))
38		fvexd 6880	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → ((𝑌(2nd ‘𝐿)((1st ‘𝐺)‘𝑧))‘𝑛) ∈ V)
39	3	fveq2d 6869	. . . . . . . . . 10 ⊢ (𝜑 → ((1st ‘𝐿)‘((1st ‘𝐾)‘𝑋)) = ((1st ‘𝐿)‘𝑌))
40		uobeq.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐷)
41		uobeq.x	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
42	11, 40, 41, 8, 17, 20	cofid1a 49029	. . . . . . . . . 10 ⊢ (𝜑 → ((1st ‘𝐿)‘((1st ‘𝐾)‘𝑋)) = 𝑋)
43	39, 42	eqtr3d 2767	. . . . . . . . 9 ⊢ (𝜑 → ((1st ‘𝐿)‘𝑌) = 𝑋)
44	43	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → ((1st ‘𝐿)‘𝑌) = 𝑋)
45	16	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → 𝐿 ∈ ((𝐸 Full 𝐷) ∩ (𝐸 Faith 𝐷)))
46		uobeq.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
47	46, 8, 17	cofuass 17857	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐿 ∘func 𝐾) ∘func 𝐹) = (𝐿 ∘func (𝐾 ∘func 𝐹)))
48	20	oveq1d 7409	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐿 ∘func 𝐾) ∘func 𝐹) = (𝐼 ∘func 𝐹))
49	46, 11	cofulid 17858	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐼 ∘func 𝐹) = 𝐹)
50	48, 49	eqtrd 2765	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐿 ∘func 𝐾) ∘func 𝐹) = 𝐹)
51	28	oveq2d 7410	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 ∘func (𝐾 ∘func 𝐹)) = (𝐿 ∘func 𝐺))
52	47, 50, 51	3eqtr3rd 2774	. . . . . . . . 9 ⊢ (𝜑 → (𝐿 ∘func 𝐺) = 𝐹)
53	52	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → (𝐿 ∘func 𝐺) = 𝐹)
54		eqidd 2731	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → ((𝑌(2nd ‘𝐿)((1st ‘𝐺)‘𝑧))‘𝑛) = ((𝑌(2nd ‘𝐿)((1st ‘𝐺)‘𝑧))‘𝑛))
55		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛)
56	44, 45, 53, 54, 55	uptrai 49124	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → 𝑧(𝐹(𝐶 UP 𝐷)𝑋)((𝑌(2nd ‘𝐿)((1st ‘𝐺)‘𝑧))‘𝑛))
57		breq2 5119	. . . . . . 7 ⊢ (𝑚 = ((𝑌(2nd ‘𝐿)((1st ‘𝐺)‘𝑧))‘𝑛) → (𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚 ↔ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)((𝑌(2nd ‘𝐿)((1st ‘𝐺)‘𝑧))‘𝑛)))
58	38, 56, 57	spcedv 3573	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → ∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚)
59	58	exlimiv 1930	. . . . 5 ⊢ (∃𝑛(𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → ∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚)
60	37, 59	sylbir 235	. . . 4 ⊢ ((𝜑 ∧ ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → ∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚)
61	36, 60	impbida 800	. . 3 ⊢ (𝜑 → (∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚 ↔ ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛))
62		relup 49090	. . . 4 ⊢ Rel (𝐹(𝐶 UP 𝐷)𝑋)
63		releldmb 5918	. . . 4 ⊢ (Rel (𝐹(𝐶 UP 𝐷)𝑋) → (𝑧 ∈ dom (𝐹(𝐶 UP 𝐷)𝑋) ↔ ∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚))
64	62, 63	ax-mp 5	. . 3 ⊢ (𝑧 ∈ dom (𝐹(𝐶 UP 𝐷)𝑋) ↔ ∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚)
65		relup 49090	. . . 4 ⊢ Rel (𝐺(𝐶 UP 𝐸)𝑌)
66		releldmb 5918	. . . 4 ⊢ (Rel (𝐺(𝐶 UP 𝐸)𝑌) → (𝑧 ∈ dom (𝐺(𝐶 UP 𝐸)𝑌) ↔ ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛))
67	65, 66	ax-mp 5	. . 3 ⊢ (𝑧 ∈ dom (𝐺(𝐶 UP 𝐸)𝑌) ↔ ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛)
68	61, 64, 67	3bitr4g 314	. 2 ⊢ (𝜑 → (𝑧 ∈ dom (𝐹(𝐶 UP 𝐷)𝑋) ↔ 𝑧 ∈ dom (𝐺(𝐶 UP 𝐸)𝑌)))
69	68	eqrdv 2728	1 ⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3455   ∩ cin 3921  ⟨cop 4603   class class class wbr 5115  dom cdm 5646  Rel wrel 5651  ‘cfv 6519  (class class class)co 7394  1^st c1st 7975  2^nd c2nd 7976  Basecbs 17185   Func cfunc 17822  id_funccidfu 17823   ∘_func ccofu 17824   Full cful 17872   Faith cfth 17873   UP cup 49081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-rmo 3357  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-riota 7351  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7977  df-2nd 7978  df-map 8805  df-ixp 8875  df-cat 17635  df-cid 17636  df-func 17826  df-idfu 17827  df-cofu 17828  df-full 17874  df-fth 17875  df-up 49082

This theorem is referenced by: (None)