Theorem uptrai 49714

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 481	. . . 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 481	. . . 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 481	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → (𝐾 ∘func 𝐹) = 𝐺)
8		eqid 2740	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
9		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀)
10	9	up1st2nd 49682	. . . . 5 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → 𝑍(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶 UP 𝐷)𝑋)𝑀)
11	10, 8	uprcl3 49687	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → 𝑋 ∈ (Base‘𝐷))
12	9	uprcl2a 49700	. . . 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 481	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑀) = 𝑁)
15		eqid 2740	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
16	10, 15	uprcl5 49689	. . . 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 49712	. . 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 693	. 2 ⊢ (𝜑 → (𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁))
19	1, 18	mpbid 233	1 ⊢ (𝜑 → 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ∩ cin 3889   class class class wbr 5079  ‘cfv 6492  (class class class)co 7363  1^st c1st 7936  2^nd c2nd 7937  Basecbs 17177  Hom chom 17229   ∘_func ccofu 17821   Full cful 17869   Faith cfth 17870   UP cup 49670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-map 8772  df-ixp 8843  df-cat 17632  df-cid 17633  df-func 17823  df-cofu 17825  df-full 17871  df-fth 17872  df-up 49671

This theorem is referenced by:  uobffth  49715  uobeqw  49716