Theorem uobffth 49197

Description: A fully faithful functor generates equal sets of universal objects. (Contributed by Zhi Wang, 19-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)
uobffth.k	⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))

Assertion

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

Proof of Theorem uobffth

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 6875	. . . . . . 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		uobffth.k	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
6	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
7		uobffth.g	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)
8	7	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → (𝐾 ∘func 𝐹) = 𝐺)
9		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))
10		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚)
11	4, 6, 8, 9, 10	uptrai 49196	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → 𝑧(𝐺(𝐶 UP 𝐸)𝑌)((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑧))‘𝑚))
12		breq2 5113	. . . . . . 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)))
13	2, 11, 12	spcedv 3567	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛)
14	13	exlimiv 1930	. . . . 5 ⊢ (∃𝑚(𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛)
15	1, 14	sylbir 235	. . . 4 ⊢ ((𝜑 ∧ ∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛)
16		19.42v 1953	. . . . 5 ⊢ (∃𝑛(𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) ↔ (𝜑 ∧ ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛))
17		fvexd 6875	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → (◡(𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑧))‘𝑛) ∈ V)
18	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → ((1st ‘𝐾)‘𝑋) = 𝑌)
19	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
20	7	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → (𝐾 ∘func 𝐹) = 𝐺)
21		uobffth.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐷)
22		uobffth.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
23	22	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → 𝑋 ∈ 𝐵)
24		uobffth.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
25	24	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → 𝐹 ∈ (𝐶 Func 𝐷))
26		eqidd 2731	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → (◡(𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑧))‘𝑛) = (◡(𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑧))‘𝑛))
27		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛)
28	18, 19, 20, 21, 23, 25, 26, 27	uptrar 49195	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → 𝑧(𝐹(𝐶 UP 𝐷)𝑋)(◡(𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑧))‘𝑛))
29		breq2 5113	. . . . . . 7 ⊢ (𝑚 = (◡(𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑧))‘𝑛) → (𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚 ↔ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)(◡(𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑧))‘𝑛)))
30	17, 28, 29	spcedv 3567	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → ∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚)
31	30	exlimiv 1930	. . . . 5 ⊢ (∃𝑛(𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → ∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚)
32	16, 31	sylbir 235	. . . 4 ⊢ ((𝜑 ∧ ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → ∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚)
33	15, 32	impbida 800	. . 3 ⊢ (𝜑 → (∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚 ↔ ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛))
34		relup 49162	. . . 4 ⊢ Rel (𝐹(𝐶 UP 𝐷)𝑋)
35		releldmb 5912	. . . 4 ⊢ (Rel (𝐹(𝐶 UP 𝐷)𝑋) → (𝑧 ∈ dom (𝐹(𝐶 UP 𝐷)𝑋) ↔ ∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚))
36	34, 35	ax-mp 5	. . 3 ⊢ (𝑧 ∈ dom (𝐹(𝐶 UP 𝐷)𝑋) ↔ ∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚)
37		relup 49162	. . . 4 ⊢ Rel (𝐺(𝐶 UP 𝐸)𝑌)
38		releldmb 5912	. . . 4 ⊢ (Rel (𝐺(𝐶 UP 𝐸)𝑌) → (𝑧 ∈ dom (𝐺(𝐶 UP 𝐸)𝑌) ↔ ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛))
39	37, 38	ax-mp 5	. . 3 ⊢ (𝑧 ∈ dom (𝐺(𝐶 UP 𝐸)𝑌) ↔ ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛)
40	33, 36, 39	3bitr4g 314	. 2 ⊢ (𝜑 → (𝑧 ∈ dom (𝐹(𝐶 UP 𝐷)𝑋) ↔ 𝑧 ∈ dom (𝐺(𝐶 UP 𝐸)𝑌)))
41	40	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 3450   ∩ cin 3915   class class class wbr 5109  ^◡ccnv 5639  dom cdm 5640  Rel wrel 5645  ‘cfv 6513  (class class class)co 7389  1^st c1st 7968  2^nd c2nd 7969  Basecbs 17185   Func cfunc 17822   ∘_func ccofu 17824   Full cful 17872   Faith cfth 17873   UP cup 49152

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 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

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 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-1st 7970  df-2nd 7971  df-map 8803  df-ixp 8873  df-cat 17635  df-cid 17636  df-func 17826  df-cofu 17828  df-full 17874  df-fth 17875  df-up 49153

This theorem is referenced by:  uobeq  49199  uobeq3  49381