Theorem uptra 49845

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 17955	. . . . 5 ⊢ Rel (𝐷 Full 𝐸)
3		relin1 5789	. . . . 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 8027	. . . 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 4191	. . . . . . 7 ⊢ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ⊆ (𝐷 Full 𝐸)
11		fullfunc 17953	. . . . . . 7 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
12	10, 11	sstri 3948	. . . . . 6 ⊢ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ⊆ (𝐷 Func 𝐸)
13	12, 5	sselid 3937	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
14	9, 13	cofu1st2nd 49722	. . . 4 ⊢ (𝜑 → (𝐾 ∘func 𝐹) = (⟨(1st ‘𝐾), (2nd ‘𝐾)⟩ ∘func ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))
15		relfunc 17907	. . . . 5 ⊢ Rel (𝐶 Func 𝐸)
16	9, 13	cofucl 17933	. . . . . 6 ⊢ (𝜑 → (𝐾 ∘func 𝐹) ∈ (𝐶 Func 𝐸))
17	8, 16	eqeltrrd 2866	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))
18		1st2nd 8024	. . . . 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 2808	. . 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 49706	. . 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 49843	. 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 49817	. 2 ⊢ (𝜑 → (𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶 UP 𝐷)𝑋)𝑀))
29	17	up1st2ndb 49817	. 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 1563   ∈ wcel 2145   ∩ cin 3906  ⟨cop 4591   class class class wbr 5104  Rel wrel 5656  ‘cfv 6525  (class class class)co 7400  1^st c1st 7972  2^nd c2nd 7973  Basecbs 17257  Hom chom 17309   Func cfunc 17899   ∘_func ccofu 17901   Full cful 17949   Faith cfth 17950   UP cup 49803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-map 8814  df-ixp 8884  df-cat 17712  df-cid 17713  df-func 17903  df-cofu 17905  df-full 17951  df-fth 17952  df-up 49804

This theorem is referenced by:  uptrar  49846  uptrai  49847