uptrai - Mathbox for Zhi Wang

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Theorem uptrai 50427

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 486	. . . 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 486	. . . 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 486	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → (𝐾 ∘_func 𝐹) = 𝐺)
8		eqid 2760	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
9		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀)
10	9	up1st2nd 50395	. . . . 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 50400	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → 𝑋 ∈ (Base‘𝐷))
12	9	uprcl2a 50413	. . . 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 486	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → ((𝑋(2^nd ‘𝐾)((1^st ‘𝐹)‘𝑍))‘𝑀) = 𝑁)
15		eqid 2760	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
16	10, 15	uprcl5 50402	. . . 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 50425	. . 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 700	. 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)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ∩ cin 4089 class class class wbr 5102 ‘cfv 6539 (class class class)co 7421 1^st c1st 8000 2^nd c2nd 8001 Basecbs 17393 Hom chom 17445 ∘_func ccofu 18037 Full cful 18085 Faith cfth 18086 UP cup 50383

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-1st 8002 df-2nd 8003 df-map 8853 df-ixp 8930 df-cat 17848 df-cid 17849 df-func 18039 df-cofu 18041 df-full 18087 df-fth 18088 df-up 50384

This theorem is used by: uobffth 50428 uobeqw 50429

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