Theorem uobffth 50665

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 1988	. . . . 5 ⊢ (∃𝑚(𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) ↔ (𝜑 ∧ ∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚))
2		fvexd 6931	. . . . . . 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 486	. . . . . . . 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 486	. . . . . . . 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 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → (𝐾 ∘func 𝐹) = 𝐺)
9		eqidd 2764	. . . . . . . 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 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚)
11	4, 6, 8, 9, 10	uptrai 50664	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → 𝑧(𝐺(𝐶 UP 𝐸)𝑌)((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑧))‘𝑚))
12		breq2 5112	. . . . . . 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 3556	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛)
14	13	exlimiv 1965	. . . . 5 ⊢ (∃𝑚(𝜑 ∧ 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛)
15	1, 14	sylbir 238	. . . 4 ⊢ ((𝜑 ∧ ∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚) → ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛)
16		19.42v 1988	. . . . 5 ⊢ (∃𝑛(𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) ↔ (𝜑 ∧ ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛))
17		fvexd 6931	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → (◡(𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑧))‘𝑛) ∈ V)
18	3	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → ((1st ‘𝐾)‘𝑋) = 𝑌)
19	5	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
20	7	adantr 486	. . . . . . . 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 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → 𝑋 ∈ 𝐵)
24		uobffth.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
25	24	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → 𝐹 ∈ (𝐶 Func 𝐷))
26		eqidd 2764	. . . . . . . 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 490	. . . . . . . 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 50663	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → 𝑧(𝐹(𝐶 UP 𝐷)𝑋)(◡(𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑧))‘𝑛))
29		breq2 5112	. . . . . . 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 3556	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → ∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚)
31	30	exlimiv 1965	. . . . 5 ⊢ (∃𝑛(𝜑 ∧ 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → ∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚)
32	16, 31	sylbir 238	. . . 4 ⊢ ((𝜑 ∧ ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛) → ∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚)
33	15, 32	impbida 813	. . 3 ⊢ (𝜑 → (∃𝑚 𝑧(𝐹(𝐶 UP 𝐷)𝑋)𝑚 ↔ ∃𝑛 𝑧(𝐺(𝐶 UP 𝐸)𝑌)𝑛))
34		relup 50630	. . . 4 ⊢ Rel (𝐹(𝐶 UP 𝐷)𝑋)
35		releldmb 5940	. . . 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 50630	. . . 4 ⊢ Rel (𝐺(𝐶 UP 𝐸)𝑌)
38		releldmb 5940	. . . 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 317	. 2 ⊢ (𝜑 → (𝑧 ∈ dom (𝐹(𝐶 UP 𝐷)𝑋) ↔ 𝑧 ∈ dom (𝐺(𝐶 UP 𝐸)𝑌)))
41	40	eqrdv 2761	1 ⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  Vcvv 3454   ∩ cin 4092   class class class wbr 5108  ^◡ccnv 5657  dom cdm 5658  Rel wrel 5663  ‘cfv 6566  (class class class)co 7451  1^st c1st 8033  2^nd c2nd 8034  Basecbs 17431   Func cfunc 18079   ∘_func ccofu 18081   Full cful 18129   Faith cfth 18130   UP cup 50620

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-map 8887  df-ixp 8964  df-cat 17892  df-cid 17893  df-func 18083  df-cofu 18085  df-full 18131  df-fth 18132  df-up 50621

This theorem is used by:  uobeq  50667  uobeq3  50868