Theorem uptr2a 49409

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 17832	. . . . 5 ⊢ Rel (𝐶 Full 𝐷)
6		relin1 5759	. . . . 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 7984	. . . 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 4187	. . . . . . 7 ⊢ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)) ⊆ (𝐶 Full 𝐷)
13		fullfunc 17830	. . . . . . 7 ⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
14	12, 13	sstri 3941	. . . . . 6 ⊢ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)) ⊆ (𝐶 Func 𝐷)
15	14, 8	sselid 3929	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝐶 Func 𝐷))
16		uptr2a.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
17	15, 16	cofu1st2nd 49279	. . . 4 ⊢ (𝜑 → (𝐺 ∘func 𝐾) = (⟨(1st ‘𝐺), (2nd ‘𝐺)⟩ ∘func ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩))
18		relfunc 17784	. . . . 5 ⊢ Rel (𝐶 Func 𝐸)
19	15, 16	cofucl 17810	. . . . . 6 ⊢ (𝜑 → (𝐺 ∘func 𝐾) ∈ (𝐶 Func 𝐸))
20	11, 19	eqeltrrd 2835	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐸))
21		1st2nd 7981	. . . . 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 2777	. . 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 49263	. . 3 ⊢ (𝜑 → (1st ‘𝐺)(𝐷 Func 𝐸)(2nd ‘𝐺))
26	1, 2, 3, 4, 10, 23, 24, 25	uptr2 49408	. 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 49374	. 2 ⊢ (𝜑 → (𝑋(𝐹(𝐶 UP 𝐸)𝑍)𝑀 ↔ 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶 UP 𝐸)𝑍)𝑀))
28	16	up1st2ndb 49374	. 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 1541   ∈ wcel 2113   ∩ cin 3898  ⟨cop 4584   class class class wbr 5096  Rel wrel 5627  –onto→wfo 6488  ‘cfv 6490  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  Basecbs 17134   Func cfunc 17776   ∘_func ccofu 17778   Full cful 17826   Faith cfth 17827   UP cup 49360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8763  df-ixp 8834  df-cat 17589  df-cid 17590  df-func 17780  df-cofu 17782  df-full 17828  df-fth 17829  df-up 49361

This theorem is referenced by:  lmdran  49858  cmdlan  49859