Theorem uptra 50546

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 18117	. . . . 5 ⊢ Rel (𝐷 Full 𝐸)
3		relin1 5796	. . . . 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 8080	. . . 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 599	. . 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 4136	. . . . . . 7 ⊢ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ⊆ (𝐷 Full 𝐸)
11		fullfunc 18115	. . . . . . 7 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
12	10, 11	sstri 3922	. . . . . 6 ⊢ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ⊆ (𝐷 Func 𝐸)
13	12, 5	sselid 3911	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
14	9, 13	cofu1st2nd 50422	. . . 4 ⊢ (𝜑 → (𝐾 ∘func 𝐹) = (⟨(1st ‘𝐾), (2nd ‘𝐾)⟩ ∘func ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))
15		relfunc 18069	. . . . 5 ⊢ Rel (𝐶 Func 𝐸)
16	9, 13	cofucl 18095	. . . . . 6 ⊢ (𝜑 → (𝐾 ∘func 𝐹) ∈ (𝐶 Func 𝐸))
17	8, 16	eqeltrrd 2865	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))
18		1st2nd 8077	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸)) → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
19	15, 17, 18	sylancr 599	. . . 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 50405	. . 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 50544	. 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 50518	. 2 ⊢ (𝜑 → (𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶 UP 𝐷)𝑋)𝑀))
29	17	up1st2ndb 50518	. 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147   ∩ cin 4092  ⟨cop 4838   class class class wbr 5106  Rel wrel 5661  ‘cfv 6561  (class class class)co 7446  1^st c1st 8026  2^nd c2nd 8027  Basecbs 17419  Hom chom 17471   Func cfunc 18061   ∘_func ccofu 18063   Full cful 18111   Faith cfth 18112   UP cup 50504

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-map 8880  df-ixp 8957  df-cat 17874  df-cid 17875  df-func 18065  df-cofu 18067  df-full 18113  df-fth 18114  df-up 50505

This theorem is used by:  uptrar  50547  uptrai  50548