Theorem uptrar 49123

Description: Universal property and fully faithful functor. (Contributed by Zhi Wang, 17-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)
uptra.b	⊢ 𝐵 = (Base‘𝐷)
uptra.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
uptra.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
uptrar.m	⊢ (𝜑 → (◡(𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑁) = 𝑀)
uptrar.z	⊢ (𝜑 → 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁)

Assertion

Ref	Expression
uptrar	⊢ (𝜑 → 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀)

Proof of Theorem uptrar

Step	Hyp	Ref	Expression
1		uptrar.z	. 2 ⊢ (𝜑 → 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁)
2		uptra.y	. . . . 5 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)
3	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → ((1st ‘𝐾)‘𝑋) = 𝑌)
4		uptra.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
5	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
6		uptra.g	. . . . 5 ⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → (𝐾 ∘func 𝐹) = 𝐺)
8		uptra.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
9		uptra.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
10	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → 𝑋 ∈ 𝐵)
11		uptra.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
12	11	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → 𝐹 ∈ (𝐶 Func 𝐷))
13		uptrar.m	. . . . . . 7 ⊢ (𝜑 → (◡(𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑁) = 𝑀)
14	13	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → (◡(𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑁) = 𝑀)
15	14	fveq2d 6869	. . . . 5 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘(◡(𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑁)) = ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑀))
16		eqid 2730	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
17		eqid 2730	. . . . . . . 8 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
18		relfull 17878	. . . . . . . . . . 11 ⊢ Rel (𝐷 Full 𝐸)
19		relin1 5783	. . . . . . . . . . 11 ⊢ (Rel (𝐷 Full 𝐸) → Rel ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
20	18, 19	ax-mp 5	. . . . . . . . . 10 ⊢ Rel ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))
21		1st2ndbr 8030	. . . . . . . . . 10 ⊢ ((Rel ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ∧ 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))) → (1st ‘𝐾)((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))(2nd ‘𝐾))
22	20, 4, 21	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝐾)((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))(2nd ‘𝐾))
23	22	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → (1st ‘𝐾)((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))(2nd ‘𝐾))
24		eqid 2730	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
25	12	func1st2nd 48993	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
26	24, 8, 25	funcf1 17834	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → (1st ‘𝐹):(Base‘𝐶)⟶𝐵)
27		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁)
28	27	up1st2nd 49092	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → 𝑍(⟨(1st ‘𝐺), (2nd ‘𝐺)⟩(𝐶 UP 𝐸)𝑌)𝑁)
29	28, 24	uprcl4 49098	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → 𝑍 ∈ (Base‘𝐶))
30	26, 29	ffvelcdmd 7064	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → ((1st ‘𝐹)‘𝑍) ∈ 𝐵)
31	8, 16, 17, 23, 10, 30	ffthf1o 17889	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → (𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍)):(𝑋(Hom ‘𝐷)((1st ‘𝐹)‘𝑍))–1-1-onto→(((1st ‘𝐾)‘𝑋)(Hom ‘𝐸)((1st ‘𝐾)‘((1st ‘𝐹)‘𝑍))))
32		inss1 4208	. . . . . . . . . . . . . 14 ⊢ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ⊆ (𝐷 Full 𝐸)
33		fullfunc 17876	. . . . . . . . . . . . . 14 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
34	32, 33	sstri 3964	. . . . . . . . . . . . 13 ⊢ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ⊆ (𝐷 Func 𝐸)
35	34, 4	sselid 3952	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
36	35	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → 𝐾 ∈ (𝐷 Func 𝐸))
37	24, 12, 36, 29	cofu1 17852	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → ((1st ‘(𝐾 ∘func 𝐹))‘𝑍) = ((1st ‘𝐾)‘((1st ‘𝐹)‘𝑍)))
38	7	fveq2d 6869	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → (1st ‘(𝐾 ∘func 𝐹)) = (1st ‘𝐺))
39	38	fveq1d 6867	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → ((1st ‘(𝐾 ∘func 𝐹))‘𝑍) = ((1st ‘𝐺)‘𝑍))
40	37, 39	eqtr3d 2767	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → ((1st ‘𝐾)‘((1st ‘𝐹)‘𝑍)) = ((1st ‘𝐺)‘𝑍))
41	3, 40	oveq12d 7412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → (((1st ‘𝐾)‘𝑋)(Hom ‘𝐸)((1st ‘𝐾)‘((1st ‘𝐹)‘𝑍))) = (𝑌(Hom ‘𝐸)((1st ‘𝐺)‘𝑍)))
42	41	f1oeq3d 6804	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍)):(𝑋(Hom ‘𝐷)((1st ‘𝐹)‘𝑍))–1-1-onto→(((1st ‘𝐾)‘𝑋)(Hom ‘𝐸)((1st ‘𝐾)‘((1st ‘𝐹)‘𝑍))) ↔ (𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍)):(𝑋(Hom ‘𝐷)((1st ‘𝐹)‘𝑍))–1-1-onto→(𝑌(Hom ‘𝐸)((1st ‘𝐺)‘𝑍))))
43	31, 42	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → (𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍)):(𝑋(Hom ‘𝐷)((1st ‘𝐹)‘𝑍))–1-1-onto→(𝑌(Hom ‘𝐸)((1st ‘𝐺)‘𝑍)))
44	28, 17	uprcl5 49099	. . . . . 6 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → 𝑁 ∈ (𝑌(Hom ‘𝐸)((1st ‘𝐺)‘𝑍)))
45		f1ocnvfv2 7259	. . . . . 6 ⊢ (((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍)):(𝑋(Hom ‘𝐷)((1st ‘𝐹)‘𝑍))–1-1-onto→(𝑌(Hom ‘𝐸)((1st ‘𝐺)‘𝑍)) ∧ 𝑁 ∈ (𝑌(Hom ‘𝐸)((1st ‘𝐺)‘𝑍))) → ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘(◡(𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑁)) = 𝑁)
46	43, 44, 45	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘(◡(𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑁)) = 𝑁)
47	15, 46	eqtr3d 2767	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑀) = 𝑁)
48		f1ocnvdm 7267	. . . . . 6 ⊢ (((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍)):(𝑋(Hom ‘𝐷)((1st ‘𝐹)‘𝑍))–1-1-onto→(𝑌(Hom ‘𝐸)((1st ‘𝐺)‘𝑍)) ∧ 𝑁 ∈ (𝑌(Hom ‘𝐸)((1st ‘𝐺)‘𝑍))) → (◡(𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑁) ∈ (𝑋(Hom ‘𝐷)((1st ‘𝐹)‘𝑍)))
49	43, 44, 48	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → (◡(𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑁) ∈ (𝑋(Hom ‘𝐷)((1st ‘𝐹)‘𝑍)))
50	14, 49	eqeltrrd 2830	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → 𝑀 ∈ (𝑋(Hom ‘𝐷)((1st ‘𝐹)‘𝑍)))
51	3, 5, 7, 8, 10, 12, 47, 16, 50	uptra 49122	. . 3 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → (𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁))
52	1, 51	mpdan 687	. 2 ⊢ (𝜑 → (𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁))
53	1, 52	mpbird 257	1 ⊢ (𝜑 → 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∩ cin 3921   class class class wbr 5115  ^◡ccnv 5645  Rel wrel 5651  –1-1-onto→wf1o 6518  ‘cfv 6519  (class class class)co 7394  1^st c1st 7975  2^nd c2nd 7976  Basecbs 17185  Hom chom 17237   Func cfunc 17822   ∘_func ccofu 17824   Full cful 17872   Faith cfth 17873   UP cup 49081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-rmo 3357  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-riota 7351  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7977  df-2nd 7978  df-map 8805  df-ixp 8875  df-cat 17635  df-cid 17636  df-func 17826  df-cofu 17828  df-full 17874  df-fth 17875  df-up 49082

This theorem is referenced by:  uobeq  49126