uptrai - Mathbox for Zhi Wang

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Theorem uptrai 49332

Description: Universal property and fully faithful functor. (Contributed by Zhi Wang, 16-Nov-2025.)

**Hypotheses**
Ref	Expression
uptra.y	⊢ (𝜑 → ((1^st ‘𝐾)‘𝑋) = 𝑌)
uptra.k	⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
uptra.g	⊢ (𝜑 → (𝐾 ∘_func 𝐹) = 𝐺)
uptrai.n	⊢ (𝜑 → ((𝑋(2^nd ‘𝐾)((1^st ‘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 → ((1^st ‘𝐾)‘𝑋) = 𝑌)
3	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → ((1^st ‘𝐾)‘𝑋) = 𝑌)
4		uptra.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
5	4	adantr 480	. . . 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 480	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → (𝐾 ∘_func 𝐹) = 𝐺)
8		eqid 2733	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
9		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀)
10	9	up1st2nd 49300	. . . . 5 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → 𝑍(⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩(𝐶 UP 𝐷)𝑋)𝑀)
11	10, 8	uprcl3 49305	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → 𝑋 ∈ (Base‘𝐷))
12	9	uprcl2a 49318	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → 𝐹 ∈ (𝐶 Func 𝐷))
13		uptrai.n	. . . . 5 ⊢ (𝜑 → ((𝑋(2^nd ‘𝐾)((1^st ‘𝐹)‘𝑍))‘𝑀) = 𝑁)
14	13	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → ((𝑋(2^nd ‘𝐾)((1^st ‘𝐹)‘𝑍))‘𝑀) = 𝑁)
15		eqid 2733	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
16	10, 15	uprcl5 49307	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → 𝑀 ∈ (𝑋(Hom ‘𝐷)((1^st ‘𝐹)‘𝑍)))
17	3, 5, 7, 8, 11, 12, 14, 15, 16	uptra 49330	. . 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 687	. 2 ⊢ (𝜑 → (𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁))
19	1, 18	mpbid 232	1 ⊢ (𝜑 → 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∩ cin 3898 class class class wbr 5095 ‘cfv 6489 (class class class)co 7355 1^st c1st 7928 2^nd c2nd 7929 Basecbs 17130 Hom chom 17182 ∘_func ccofu 17773 Full cful 17821 Faith cfth 17822 UP cup 49288

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 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677

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 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-1st 7930 df-2nd 7931 df-map 8761 df-ixp 8831 df-cat 17584 df-cid 17585 df-func 17775 df-cofu 17777 df-full 17823 df-fth 17824 df-up 49289

This theorem is referenced by: uobffth 49333 uobeqw 49334

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