Theorem uptra 49648

Description: Universal property and fully faithful functor. (Contributed by Zhi Wang, 16-Nov-2025.)

Hypotheses

Ref	Expression
uptra.y	⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)
uptra.k	⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
uptra.g	⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)
uptra.b	⊢ 𝐵 = (Base‘𝐷)
uptra.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
uptra.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
uptra.n	⊢ (𝜑 → ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑀) = 𝑁)
uptra.j	⊢ 𝐽 = (Hom ‘𝐷)
uptra.m	⊢ (𝜑 → 𝑀 ∈ (𝑋𝐽((1st ‘𝐹)‘𝑍)))

Assertion

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

Proof of Theorem uptra

Step	Hyp	Ref	Expression
1		uptra.y	. . 3 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)
2		relfull 17835	. . . . 5 ⊢ Rel (𝐷 Full 𝐸)
3		relin1 5759	. . . . 5 ⊢ (Rel (𝐷 Full 𝐸) → Rel ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
4	2, 3	ax-mp 5	. . . 4 ⊢ Rel ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))
5		uptra.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
6		1st2ndbr 7986	. . . 4 ⊢ ((Rel ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ∧ 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))) → (1st ‘𝐾)((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))(2nd ‘𝐾))
7	4, 5, 6	sylancr 588	. . 3 ⊢ (𝜑 → (1st ‘𝐾)((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))(2nd ‘𝐾))
8		uptra.g	. . . 4 ⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)
9		uptra.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
10		inss1 4178	. . . . . . 7 ⊢ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ⊆ (𝐷 Full 𝐸)
11		fullfunc 17833	. . . . . . 7 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
12	10, 11	sstri 3932	. . . . . 6 ⊢ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ⊆ (𝐷 Func 𝐸)
13	12, 5	sselid 3920	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
14	9, 13	cofu1st2nd 49525	. . . 4 ⊢ (𝜑 → (𝐾 ∘func 𝐹) = (⟨(1st ‘𝐾), (2nd ‘𝐾)⟩ ∘func ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))
15		relfunc 17787	. . . . 5 ⊢ Rel (𝐶 Func 𝐸)
16	9, 13	cofucl 17813	. . . . . 6 ⊢ (𝜑 → (𝐾 ∘func 𝐹) ∈ (𝐶 Func 𝐸))
17	8, 16	eqeltrrd 2838	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))
18		1st2nd 7983	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸)) → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
19	15, 17, 18	sylancr 588	. . . 4 ⊢ (𝜑 → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
20	8, 14, 19	3eqtr3d 2780	. . 3 ⊢ (𝜑 → (⟨(1st ‘𝐾), (2nd ‘𝐾)⟩ ∘func ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩) = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
21		uptra.b	. . 3 ⊢ 𝐵 = (Base‘𝐷)
22		uptra.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
23	9	func1st2nd 49509	. . 3 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
24		uptra.n	. . 3 ⊢ (𝜑 → ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑀) = 𝑁)
25		uptra.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐷)
26		uptra.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐽((1st ‘𝐹)‘𝑍)))
27	1, 7, 20, 21, 22, 23, 24, 25, 26	uptr 49646	. 2 ⊢ (𝜑 → (𝑍(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(⟨(1st ‘𝐺), (2nd ‘𝐺)⟩(𝐶 UP 𝐸)𝑌)𝑁))
28	9	up1st2ndb 49620	. 2 ⊢ (𝜑 → (𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶 UP 𝐷)𝑋)𝑀))
29	17	up1st2ndb 49620	. 2 ⊢ (𝜑 → (𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁 ↔ 𝑍(⟨(1st ‘𝐺), (2nd ‘𝐺)⟩(𝐶 UP 𝐸)𝑌)𝑁))
30	27, 28, 29	3bitr4d 311	1 ⊢ (𝜑 → (𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114   ∩ cin 3889  ⟨cop 4574   class class class wbr 5086  Rel wrel 5627  ‘cfv 6490  (class class class)co 7358  1^st c1st 7931  2^nd c2nd 7932  Basecbs 17137  Hom chom 17189   Func cfunc 17779   ∘_func ccofu 17781   Full cful 17829   Faith cfth 17830   UP cup 49606

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

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 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  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 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8766  df-ixp 8837  df-cat 17592  df-cid 17593  df-func 17783  df-cofu 17785  df-full 17831  df-fth 17832  df-up 49607

This theorem is referenced by:  uptrar  49649  uptrai  49650