Theorem uptr2a 49585

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 17846	. . . . 5 ⊢ Rel (𝐶 Full 𝐷)
6		relin1 5769	. . . . 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 7996	. . . 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 588	. . 3 ⊢ (𝜑 → (1st ‘𝐾)((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))(2nd ‘𝐾))
11		uptr2a.f	. . . 4 ⊢ (𝜑 → (𝐺 ∘func 𝐾) = 𝐹)
12		inss1 4191	. . . . . . 7 ⊢ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)) ⊆ (𝐶 Full 𝐷)
13		fullfunc 17844	. . . . . . 7 ⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
14	12, 13	sstri 3945	. . . . . 6 ⊢ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)) ⊆ (𝐶 Func 𝐷)
15	14, 8	sselid 3933	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝐶 Func 𝐷))
16		uptr2a.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
17	15, 16	cofu1st2nd 49455	. . . 4 ⊢ (𝜑 → (𝐺 ∘func 𝐾) = (⟨(1st ‘𝐺), (2nd ‘𝐺)⟩ ∘func ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩))
18		relfunc 17798	. . . . 5 ⊢ Rel (𝐶 Func 𝐸)
19	15, 16	cofucl 17824	. . . . . 6 ⊢ (𝜑 → (𝐺 ∘func 𝐾) ∈ (𝐶 Func 𝐸))
20	11, 19	eqeltrrd 2838	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐸))
21		1st2nd 7993	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐸) ∧ 𝐹 ∈ (𝐶 Func 𝐸)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
22	18, 20, 21	sylancr 588	. . . 4 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
23	11, 17, 22	3eqtr3d 2780	. . 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 49439	. . 3 ⊢ (𝜑 → (1st ‘𝐺)(𝐷 Func 𝐸)(2nd ‘𝐺))
26	1, 2, 3, 4, 10, 23, 24, 25	uptr2 49584	. 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 49550	. 2 ⊢ (𝜑 → (𝑋(𝐹(𝐶 UP 𝐸)𝑍)𝑀 ↔ 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶 UP 𝐸)𝑍)𝑀))
28	16	up1st2ndb 49550	. 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 1542   ∈ wcel 2114   ∩ cin 3902  ⟨cop 4588   class class class wbr 5100  Rel wrel 5637  –onto→wfo 6498  ‘cfv 6500  (class class class)co 7368  1^st c1st 7941  2^nd c2nd 7942  Basecbs 17148   Func cfunc 17790   ∘_func ccofu 17792   Full cful 17840   Faith cfth 17841   UP cup 49536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-map 8777  df-ixp 8848  df-cat 17603  df-cid 17604  df-func 17794  df-cofu 17796  df-full 17842  df-fth 17843  df-up 49537

This theorem is referenced by:  lmdran  50034  cmdlan  50035