Theorem uptr2a 49488

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 17836	. . . . 5 ⊢ Rel (𝐶 Full 𝐷)
6		relin1 5761	. . . . 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 7986	. . . 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 4189	. . . . . . 7 ⊢ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)) ⊆ (𝐶 Full 𝐷)
13		fullfunc 17834	. . . . . . 7 ⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
14	12, 13	sstri 3943	. . . . . 6 ⊢ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)) ⊆ (𝐶 Func 𝐷)
15	14, 8	sselid 3931	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝐶 Func 𝐷))
16		uptr2a.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
17	15, 16	cofu1st2nd 49358	. . . 4 ⊢ (𝜑 → (𝐺 ∘func 𝐾) = (⟨(1st ‘𝐺), (2nd ‘𝐺)⟩ ∘func ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩))
18		relfunc 17788	. . . . 5 ⊢ Rel (𝐶 Func 𝐸)
19	15, 16	cofucl 17814	. . . . . 6 ⊢ (𝜑 → (𝐺 ∘func 𝐾) ∈ (𝐶 Func 𝐸))
20	11, 19	eqeltrrd 2837	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐸))
21		1st2nd 7983	. . . . 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 2779	. . 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 49342	. . 3 ⊢ (𝜑 → (1st ‘𝐺)(𝐷 Func 𝐸)(2nd ‘𝐺))
26	1, 2, 3, 4, 10, 23, 24, 25	uptr2 49487	. 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 49453	. 2 ⊢ (𝜑 → (𝑋(𝐹(𝐶 UP 𝐸)𝑍)𝑀 ↔ 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶 UP 𝐸)𝑍)𝑀))
28	16	up1st2ndb 49453	. 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 3900  ⟨cop 4586   class class class wbr 5098  Rel wrel 5629  –onto→wfo 6490  ‘cfv 6492  (class class class)co 7358  1^st c1st 7931  2^nd c2nd 7932  Basecbs 17138   Func cfunc 17780   ∘_func ccofu 17782   Full cful 17830   Faith cfth 17831   UP cup 49439

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8767  df-ixp 8838  df-cat 17593  df-cid 17594  df-func 17784  df-cofu 17786  df-full 17832  df-fth 17833  df-up 49440

This theorem is referenced by:  lmdran  49937  cmdlan  49938