uptrai - Mathbox for Zhi Wang

	Mathbox for Zhi Wang		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > uptrai		Structured version Visualization version GIF version

Theorem uptrai 50151

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 2762	. . . 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 50119	. . . . 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 50124	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) → 𝑋 ∈ (Base‘𝐷))
12	9	uprcl2a 50137	. . . 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 2762	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
16	10, 15	uprcl5 50126	. . . 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 50149	. . 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 1570 ∈ wcel 2145 ∩ cin 3901 class class class wbr 5107 ‘cfv 6537 (class class class)co 7417 1^st c1st 7988 2^nd c2nd 7989 Basecbs 17307 Hom chom 17359 ∘_func ccofu 17951 Full cful 17999 Faith cfth 18000 UP cup 50107

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740

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 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-1st 7990 df-2nd 7991 df-map 8832 df-ixp 8909 df-cat 17762 df-cid 17763 df-func 17953 df-cofu 17955 df-full 18001 df-fth 18002 df-up 50108

This theorem is used by: uobffth 50152 uobeqw 50153

W3C validator