Theorem uptra 49976

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 17968	. . . . 5 ⊢ Rel (𝐷 Full 𝐸)
3		relin1 5801	. . . . 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 8040	. . . 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 598	. . 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 4190	. . . . . . 7 ⊢ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ⊆ (𝐷 Full 𝐸)
11		fullfunc 17966	. . . . . . 7 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
12	10, 11	sstri 3947	. . . . . 6 ⊢ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ⊆ (𝐷 Func 𝐸)
13	12, 5	sselid 3936	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
14	9, 13	cofu1st2nd 49853	. . . 4 ⊢ (𝜑 → (𝐾 ∘func 𝐹) = (⟨(1st ‘𝐾), (2nd ‘𝐾)⟩ ∘func ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))
15		relfunc 17920	. . . . 5 ⊢ Rel (𝐶 Func 𝐸)
16	9, 13	cofucl 17946	. . . . . 6 ⊢ (𝜑 → (𝐾 ∘func 𝐹) ∈ (𝐶 Func 𝐸))
17	8, 16	eqeltrrd 2864	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))
18		1st2nd 8037	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸)) → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
19	15, 17, 18	sylancr 598	. . . 4 ⊢ (𝜑 → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
20	8, 14, 19	3eqtr3d 2806	. . 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 49837	. . 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 49974	. 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 49948	. 2 ⊢ (𝜑 → (𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶 UP 𝐷)𝑋)𝑀))
29	17	up1st2ndb 49948	. 2 ⊢ (𝜑 → (𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁 ↔ 𝑍(⟨(1st ‘𝐺), (2nd ‘𝐺)⟩(𝐶 UP 𝐸)𝑌)𝑁))
30	27, 28, 29	3bitr4d 314	1 ⊢ (𝜑 → (𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2143   ∩ cin 3905  ⟨cop 4596   class class class wbr 5110  Rel wrel 5668  ‘cfv 6538  (class class class)co 7412  1^st c1st 7985  2^nd c2nd 7986  Basecbs 17270  Hom chom 17322   Func cfunc 17912   ∘_func ccofu 17914   Full cful 17962   Faith cfth 17963   UP cup 49934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-map 8827  df-ixp 8897  df-cat 17725  df-cid 17726  df-func 17916  df-cofu 17918  df-full 17964  df-fth 17965  df-up 49935

This theorem is referenced by:  uptrar  49977  uptrai  49978