Theorem uobeqw 49706

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
uobffth.b	⊢ 𝐵 = (Base‘𝐷)
uobffth.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
uobffth.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
uobffth.g	⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)
uobffth.y	⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)
uobeq.i	⊢ 𝐼 = (idfunc‘𝐷)
uobeq.k	⊢ (𝜑 → 𝐾 ∈ (𝐷 Full 𝐸))
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 1955	. . . . 5 ⊢ (∃𝑚(𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) ↔ (𝜑 ∧ ∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚))
2		fvexd 6849	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑧))‘𝑚) ∈ V)
3		uobffth.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 17820	. . . . . . . . . . . 12 ⊢ Rel (𝐷 Func 𝐸)
7		fullfunc 17866	. . . . . . . . . . . . 13 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
8	7, 5	sselid 3920	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
9		1st2nd 7985	. . . . . . . . . . . 12 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)) → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
10	6, 8, 9	sylancr 588	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
11		uobeq.i	. . . . . . . . . . . . 13 ⊢ 𝐼 = (idfunc‘𝐷)
12	8	func1st2nd 49563	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾))
13		inss1 4178	. . . . . . . . . . . . . . . 16 ⊢ ((𝐸 Full 𝐷) ∩ (𝐸 Faith 𝐷)) ⊆ (𝐸 Full 𝐷)
14		fullfunc 17866	. . . . . . . . . . . . . . . 16 ⊢ (𝐸 Full 𝐷) ⊆ (𝐸 Func 𝐷)
15	13, 14	sstri 3932	. . . . . . . . . . . . . . 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 3920	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐿 ∈ (𝐸 Func 𝐷))
18	17	func1st2nd 49563	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1st ‘𝐿)(𝐸 Func 𝐷)(2nd ‘𝐿))
19	8, 17	cofu1st2nd 49579	. . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⟨(1st ‘𝐿), (2nd ‘𝐿)⟩ ∘func ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩) = 𝐼)
22	11, 12, 18, 21	cofidfth 49649	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Faith 𝐸)(2nd ‘𝐾))
23		df-br 5087	. . . . . . . . . . . 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 2837	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Faith 𝐸))
26	5, 25	elind 4141	. . . . . . . . 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		uobffth.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 2738	. . . . . . . 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 49704	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → 𝑧(𝐺(𝐶 UP 𝐸)𝑌)((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑧))‘𝑚))
33		breq2 5090	. . . . . . 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 3541	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛)
35	34	exlimiv 1932	. . . . 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 1955	. . . . 5 ⊢ (∃𝑛(𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) ↔ (𝜑 ∧ ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛))
38		fvexd 6849	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → ((𝑌(2nd ‘𝐿)((1st ‘𝐺)‘𝑧))‘𝑛) ∈ V)
39	3	fveq2d 6838	. . . . . . . . . 10 ⊢ (𝜑 → ((1st ‘𝐿)‘((1st ‘𝐾)‘𝑋)) = ((1st ‘𝐿)‘𝑌))
40		uobffth.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐷)
41		uobffth.x	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
42	11, 40, 41, 8, 17, 20	cofid1a 49599	. . . . . . . . . 10 ⊢ (𝜑 → ((1st ‘𝐿)‘((1st ‘𝐾)‘𝑋)) = 𝑋)
43	39, 42	eqtr3d 2774	. . . . . . . . 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		uobffth.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
47	46, 8, 17	cofuass 17847	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐿 ∘func 𝐾) ∘func 𝐹) = (𝐿 ∘func (𝐾 ∘func 𝐹)))
48	20	oveq1d 7375	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐿 ∘func 𝐾) ∘func 𝐹) = (𝐼 ∘func 𝐹))
49	46, 11	cofulid 17848	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐼 ∘func 𝐹) = 𝐹)
50	48, 49	eqtrd 2772	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐿 ∘func 𝐾) ∘func 𝐹) = 𝐹)
51	28	oveq2d 7376	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 ∘func (𝐾 ∘func 𝐹)) = (𝐿 ∘func 𝐺))
52	47, 50, 51	3eqtr3rd 2781	. . . . . . . . 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 2738	. . . . . . . 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 49704	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → 𝑧(𝐹(𝐶 UP 𝐷)𝑋)((𝑌(2nd ‘𝐿)((1st ‘𝐺)‘𝑧))‘𝑛))
57		breq2 5090	. . . . . . 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 3541	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → ∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚)
59	58	exlimiv 1932	. . . . 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 801	. . 3 ⊢ (𝜑 → (∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚 ↔ ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛))
62		relup 49670	. . . 4 ⊢ Rel (𝐹(𝐶 UP 𝐷)𝑋)
63		releldmb 5895	. . . 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 49670	. . . 4 ⊢ Rel (𝐺(𝐶 UP 𝐸)𝑌)
66		releldmb 5895	. . . 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 2735	1 ⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3430   ∩ cin 3889  ⟨cop 4574   class class class wbr 5086  dom cdm 5624  Rel wrel 5629  ‘cfv 6492  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934  Basecbs 17170   Func cfunc 17812  id_funccidfu 17813   ∘_func ccofu 17814   Full cful 17862   Faith cfth 17863   UP cup 49660

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  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 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-map 8768  df-ixp 8839  df-cat 17625  df-cid 17626  df-func 17816  df-idfu 17817  df-cofu 17818  df-full 17864  df-fth 17865  df-up 49661

This theorem is referenced by: (None)