Theorem uptrar 50268

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 486	. . . 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 486	. . . 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 486	. . . 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 486	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → 𝑋 ∈ 𝐵)
11		uptra.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
12	11	adantr 486	. . . 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 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → (◡(𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑁) = 𝑀)
15	14	fveq2d 6881	. . . . 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 2761	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
17		eqid 2761	. . . . . . . 8 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
18		relfull 18065	. . . . . . . . . . 11 ⊢ Rel (𝐷 Full 𝐸)
19		relin1 5790	. . . . . . . . . . 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 8042	. . . . . . . . . 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 599	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝐾)((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))(2nd ‘𝐾))
23	22	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → (1st ‘𝐾)((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))(2nd ‘𝐾))
24		eqid 2761	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
25	12	func1st2nd 50128	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
26	24, 8, 25	funcf1 18021	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → (1st ‘𝐹):(Base‘𝐶)⟶𝐵)
27		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁)
28	27	up1st2nd 50237	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → 𝑍(⟨(1st ‘𝐺), (2nd ‘𝐺)⟩(𝐶 UP 𝐸)𝑌)𝑁)
29	28, 24	uprcl4 50243	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → 𝑍 ∈ (Base‘𝐶))
30	26, 29	ffvelcdmd 7077	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → ((1st ‘𝐹)‘𝑍) ∈ 𝐵)
31	8, 16, 17, 23, 10, 30	ffthf1o 18076	. . . . . . 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 4182	. . . . . . . . . . . . . 14 ⊢ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ⊆ (𝐷 Full 𝐸)
33		fullfunc 18063	. . . . . . . . . . . . . 14 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
34	32, 33	sstri 3940	. . . . . . . . . . . . 13 ⊢ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ⊆ (𝐷 Func 𝐸)
35	34, 4	sselid 3929	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
36	35	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → 𝐾 ∈ (𝐷 Func 𝐸))
37	24, 12, 36, 29	cofu1 18039	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → ((1st ‘(𝐾 ∘func 𝐹))‘𝑍) = ((1st ‘𝐾)‘((1st ‘𝐹)‘𝑍)))
38	7	fveq2d 6881	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → (1st ‘(𝐾 ∘func 𝐹)) = (1st ‘𝐺))
39	38	fveq1d 6879	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → ((1st ‘(𝐾 ∘func 𝐹))‘𝑍) = ((1st ‘𝐺)‘𝑍))
40	37, 39	eqtr3d 2798	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → ((1st ‘𝐾)‘((1st ‘𝐹)‘𝑍)) = ((1st ‘𝐺)‘𝑍))
41	3, 40	oveq12d 7430	. . . . . . . 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 6813	. . . . . . 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 235	. . . . . 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 50244	. . . . . 6 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → 𝑁 ∈ (𝑌(Hom ‘𝐸)((1st ‘𝐺)‘𝑍)))
45		f1ocnvfv2 7277	. . . . . 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 596	. . . . 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 2798	. . . 4 ⊢ ((𝜑 ∧ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) → ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑀) = 𝑁)
48		f1ocnvdm 7285	. . . . . 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 596	. . . . 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 2862	. . . 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 50267	. . 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 700	. 2 ⊢ (𝜑 → (𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁))
53	1, 52	mpbird 260	1 ⊢ (𝜑 → 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∩ cin 3898   class class class wbr 5103  ^◡ccnv 5650  Rel wrel 5656  –1-1-onto→wf1o 6530  ‘cfv 6531  (class class class)co 7412  1^st c1st 7988  2^nd c2nd 7989  Basecbs 17367  Hom chom 17419   Func cfunc 18009   ∘_func ccofu 18011   Full cful 18059   Faith cfth 18060   UP cup 50225

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 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-map 8833  df-ixp 8910  df-cat 17822  df-cid 17823  df-func 18013  df-cofu 18015  df-full 18061  df-fth 18062  df-up 50226

This theorem is used by:  uobffth  50270