Theorem uptr2a 49201

Description: Universal property and fully faithful functor surjective on objects. (Contributed by Zhi Wang, 25-Nov-2025.)

Hypotheses

Ref	Expression
uptr2a.a	⊢ 𝐴 = (Base‘𝐶)
uptr2a.b	⊢ 𝐵 = (Base‘𝐷)
uptr2a.y	⊢ (𝜑 → 𝑌 = ((1st ‘𝐾)‘𝑋))
uptr2a.f	⊢ (𝜑 → (𝐺 ∘func 𝐾) = 𝐹)
uptr2a.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
uptr2a.g	⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
uptr2a.k	⊢ (𝜑 → 𝐾 ∈ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)))
uptr2a.1	⊢ (𝜑 → (1st ‘𝐾):𝐴–onto→𝐵)

Assertion

Ref	Expression
uptr2a	⊢ (𝜑 → (𝑋(𝐹(𝐶 UP 𝐸)𝑍)𝑀 ↔ 𝑌(𝐺(𝐷 UP 𝐸)𝑍)𝑀))

Proof of Theorem uptr2a

Step	Hyp	Ref	Expression
1		uptr2a.a	. . 3 ⊢ 𝐴 = (Base‘𝐶)
2		uptr2a.b	. . 3 ⊢ 𝐵 = (Base‘𝐷)
3		uptr2a.y	. . 3 ⊢ (𝜑 → 𝑌 = ((1st ‘𝐾)‘𝑋))
4		uptr2a.1	. . 3 ⊢ (𝜑 → (1st ‘𝐾):𝐴–onto→𝐵)
5		relfull 17878	. . . . 5 ⊢ Rel (𝐶 Full 𝐷)
6		relin1 5777	. . . . 5 ⊢ (Rel (𝐶 Full 𝐷) → Rel ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)))
7	5, 6	ax-mp 5	. . . 4 ⊢ Rel ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))
8		uptr2a.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)))
9		1st2ndbr 8023	. . . 4 ⊢ ((Rel ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)) ∧ 𝐾 ∈ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))) → (1st ‘𝐾)((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))(2nd ‘𝐾))
10	7, 8, 9	sylancr 587	. . 3 ⊢ (𝜑 → (1st ‘𝐾)((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))(2nd ‘𝐾))
11		uptr2a.f	. . . 4 ⊢ (𝜑 → (𝐺 ∘func 𝐾) = 𝐹)
12		inss1 4202	. . . . . . 7 ⊢ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)) ⊆ (𝐶 Full 𝐷)
13		fullfunc 17876	. . . . . . 7 ⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
14	12, 13	sstri 3958	. . . . . 6 ⊢ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)) ⊆ (𝐶 Func 𝐷)
15	14, 8	sselid 3946	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝐶 Func 𝐷))
16		uptr2a.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
17	15, 16	cofu1st2nd 49071	. . . 4 ⊢ (𝜑 → (𝐺 ∘func 𝐾) = (⟨(1st ‘𝐺), (2nd ‘𝐺)⟩ ∘func ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩))
18		relfunc 17830	. . . . 5 ⊢ Rel (𝐶 Func 𝐸)
19	15, 16	cofucl 17856	. . . . . 6 ⊢ (𝜑 → (𝐺 ∘func 𝐾) ∈ (𝐶 Func 𝐸))
20	11, 19	eqeltrrd 2830	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐸))
21		1st2nd 8020	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐸) ∧ 𝐹 ∈ (𝐶 Func 𝐸)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
22	18, 20, 21	sylancr 587	. . . 4 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
23	11, 17, 22	3eqtr3d 2773	. . 3 ⊢ (𝜑 → (⟨(1st ‘𝐺), (2nd ‘𝐺)⟩ ∘func ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩) = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
24		uptr2a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
25	16	func1st2nd 49055	. . 3 ⊢ (𝜑 → (1st ‘𝐺)(𝐷 Func 𝐸)(2nd ‘𝐺))
26	1, 2, 3, 4, 10, 23, 24, 25	uptr2 49200	. 2 ⊢ (𝜑 → (𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶 UP 𝐸)𝑍)𝑀 ↔ 𝑌(⟨(1st ‘𝐺), (2nd ‘𝐺)⟩(𝐷 UP 𝐸)𝑍)𝑀))
27	20	up1st2ndb 49166	. 2 ⊢ (𝜑 → (𝑋(𝐹(𝐶 UP 𝐸)𝑍)𝑀 ↔ 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶 UP 𝐸)𝑍)𝑀))
28	16	up1st2ndb 49166	. 2 ⊢ (𝜑 → (𝑌(𝐺(𝐷 UP 𝐸)𝑍)𝑀 ↔ 𝑌(⟨(1st ‘𝐺), (2nd ‘𝐺)⟩(𝐷 UP 𝐸)𝑍)𝑀))
29	26, 27, 28	3bitr4d 311	1 ⊢ (𝜑 → (𝑋(𝐹(𝐶 UP 𝐸)𝑍)𝑀 ↔ 𝑌(𝐺(𝐷 UP 𝐸)𝑍)𝑀))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109   ∩ cin 3915  ⟨cop 4597   class class class wbr 5109  Rel wrel 5645  –onto→wfo 6511  ‘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:  lmdran  49650  cmdlan  49651