Theorem uptrai 49846

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)
uptrai.n	⊢ (𝜑 → ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑀) = 𝑁)
uptrai.z	⊢ (𝜑 → 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀)

Assertion

Ref	Expression
uptrai	⊢ (𝜑 → 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁)

Proof of Theorem uptrai

Step	Hyp	Ref	Expression
1		uptrai.z	. 2 ⊢ (𝜑 → 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀)
2		uptra.y	. . . . 5 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)
3	2	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → ((1st ‘𝐾)‘𝑋) = 𝑌)
4		uptra.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
5	4	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
6		uptra.g	. . . . 5 ⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)
7	6	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → (𝐾 ∘func 𝐹) = 𝐺)
8		eqid 2765	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
9		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀)
10	9	up1st2nd 49814	. . . . 5 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → 𝑍(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶 UP 𝐷)𝑋)𝑀)
11	10, 8	uprcl3 49819	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → 𝑋 ∈ (Base‘𝐷))
12	9	uprcl2a 49832	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → 𝐹 ∈ (𝐶 Func 𝐷))
13		uptrai.n	. . . . 5 ⊢ (𝜑 → ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑀) = 𝑁)
14	13	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑀) = 𝑁)
15		eqid 2765	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
16	10, 15	uprcl5 49821	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → 𝑀 ∈ (𝑋(Hom ‘𝐷)((1st ‘𝐹)‘𝑍)))
17	3, 5, 7, 8, 11, 12, 14, 15, 16	uptra 49844	. . 3 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → (𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁))
18	1, 17	mpdan 699	. 2 ⊢ (𝜑 → (𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁))
19	1, 18	mpbid 235	1 ⊢ (𝜑 → 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ∩ cin 3906   class class class wbr 5105  ‘cfv 6525  (class class class)co 7400  1^st c1st 7972  2^nd c2nd 7973  Basecbs 17259  Hom chom 17311   ∘_func ccofu 17903   Full cful 17951   Faith cfth 17952   UP cup 49802

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 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  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 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  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 17714  df-cid 17715  df-func 17905  df-cofu 17907  df-full 17953  df-fth 17954  df-up 49803

This theorem is referenced by:  uobffth  49847  uobeqw  49848