Theorem uptra 49340

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 17819	. . . . 5 ⊢ Rel (𝐷 Full 𝐸)
3		relin1 5756	. . . . 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 7980	. . . 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 587	. . 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 4186	. . . . . . 7 ⊢ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ⊆ (𝐷 Full 𝐸)
11		fullfunc 17817	. . . . . . 7 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
12	10, 11	sstri 3940	. . . . . 6 ⊢ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ⊆ (𝐷 Func 𝐸)
13	12, 5	sselid 3928	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
14	9, 13	cofu1st2nd 49217	. . . 4 ⊢ (𝜑 → (𝐾 ∘func 𝐹) = (⟨(1st ‘𝐾), (2nd ‘𝐾)⟩ ∘func ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))
15		relfunc 17771	. . . . 5 ⊢ Rel (𝐶 Func 𝐸)
16	9, 13	cofucl 17797	. . . . . 6 ⊢ (𝜑 → (𝐾 ∘func 𝐹) ∈ (𝐶 Func 𝐸))
17	8, 16	eqeltrrd 2834	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))
18		1st2nd 7977	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸)) → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
19	15, 17, 18	sylancr 587	. . . 4 ⊢ (𝜑 → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
20	8, 14, 19	3eqtr3d 2776	. . 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 49201	. . 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 49338	. 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 49312	. 2 ⊢ (𝜑 → (𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶 UP 𝐷)𝑋)𝑀))
29	17	up1st2ndb 49312	. 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 1541   ∈ wcel 2113   ∩ cin 3897  ⟨cop 4581   class class class wbr 5093  Rel wrel 5624  ‘cfv 6486  (class class class)co 7352  1^st c1st 7925  2^nd c2nd 7926  Basecbs 17122  Hom chom 17174   Func cfunc 17763   ∘_func ccofu 17765   Full cful 17813   Faith cfth 17814   UP cup 49298

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 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-map 8758  df-ixp 8828  df-cat 17576  df-cid 17577  df-func 17767  df-cofu 17769  df-full 17815  df-fth 17816  df-up 49299

This theorem is referenced by:  uptrar  49341  uptrai  49342