Theorem uobeqw 49208

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 1953	. . . . 5 ⊢ (∃𝑚(𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) ↔ (𝜑 ∧ ∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚))
2		fvexd 6873	. . . . . . 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 17824	. . . . . . . . . . . 12 ⊢ Rel (𝐷 Func 𝐸)
7		fullfunc 17870	. . . . . . . . . . . . 13 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
8	7, 5	sselid 3944	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
9		1st2nd 8018	. . . . . . . . . . . 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 49065	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾))
13		inss1 4200	. . . . . . . . . . . . . . . 16 ⊢ ((𝐸 Full 𝐷) ∩ (𝐸 Faith 𝐷)) ⊆ (𝐸 Full 𝐷)
14		fullfunc 17870	. . . . . . . . . . . . . . . 16 ⊢ (𝐸 Full 𝐷) ⊆ (𝐸 Func 𝐷)
15	13, 14	sstri 3956	. . . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐿 ∈ (𝐸 Func 𝐷))
18	17	func1st2nd 49065	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1st ‘𝐿)(𝐸 Func 𝐷)(2nd ‘𝐿))
19	8, 17	cofu1st2nd 49081	. . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⟨(1st ‘𝐿), (2nd ‘𝐿)⟩ ∘func ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩) = 𝐼)
22	11, 12, 18, 21	cofidfth 49151	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Faith 𝐸)(2nd ‘𝐾))
23		df-br 5108	. . . . . . . . . . . 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 2828	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Faith 𝐸))
26	5, 25	elind 4163	. . . . . . . . 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 2730	. . . . . . . 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 49206	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → 𝑧(𝐺(𝐶 UP 𝐸)𝑌)((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑧))‘𝑚))
33		breq2 5111	. . . . . . 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 3564	. . . . . 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 6873	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → ((𝑌(2nd ‘𝐿)((1st ‘𝐺)‘𝑧))‘𝑛) ∈ V)
39	3	fveq2d 6862	. . . . . . . . . 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 49101	. . . . . . . . . 10 ⊢ (𝜑 → ((1st ‘𝐿)‘((1st ‘𝐾)‘𝑋)) = 𝑋)
43	39, 42	eqtr3d 2766	. . . . . . . . 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 17851	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐿 ∘func 𝐾) ∘func 𝐹) = (𝐿 ∘func (𝐾 ∘func 𝐹)))
48	20	oveq1d 7402	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐿 ∘func 𝐾) ∘func 𝐹) = (𝐼 ∘func 𝐹))
49	46, 11	cofulid 17852	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐼 ∘func 𝐹) = 𝐹)
50	48, 49	eqtrd 2764	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐿 ∘func 𝐾) ∘func 𝐹) = 𝐹)
51	28	oveq2d 7403	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 ∘func (𝐾 ∘func 𝐹)) = (𝐿 ∘func 𝐺))
52	47, 50, 51	3eqtr3rd 2773	. . . . . . . . 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 2730	. . . . . . . 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 49206	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → 𝑧(𝐹(𝐶 UP 𝐷)𝑋)((𝑌(2nd ‘𝐿)((1st ‘𝐺)‘𝑧))‘𝑛))
57		breq2 5111	. . . . . . 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 3564	. . . . . 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 49172	. . . 4 ⊢ Rel (𝐹(𝐶 UP 𝐷)𝑋)
63		releldmb 5910	. . . 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 49172	. . . 4 ⊢ Rel (𝐺(𝐶 UP 𝐸)𝑌)
66		releldmb 5910	. . . 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 2727	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 3447   ∩ cin 3913  ⟨cop 4595   class class class wbr 5107  dom cdm 5638  Rel wrel 5643  ‘cfv 6511  (class class class)co 7387  1^st c1st 7966  2^nd c2nd 7967  Basecbs 17179   Func cfunc 17816  id_funccidfu 17817   ∘_func ccofu 17818   Full cful 17866   Faith cfth 17867   UP cup 49162

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-map 8801  df-ixp 8871  df-cat 17629  df-cid 17630  df-func 17820  df-idfu 17821  df-cofu 17822  df-full 17868  df-fth 17869  df-up 49163

This theorem is referenced by: (None)