Theorem uptr 50265

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

Hypotheses

Ref	Expression
uptr.y	⊢ (𝜑 → (𝑅‘𝑋) = 𝑌)
uptr.r	⊢ (𝜑 → 𝑅((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))𝑆)
uptr.k	⊢ (𝜑 → (⟨𝑅, 𝑆⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝐾, 𝐿⟩)
uptr.b	⊢ 𝐵 = (Base‘𝐷)
uptr.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
uptr.f	⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
uptr.n	⊢ (𝜑 → ((𝑋𝑆(𝐹‘𝑍))‘𝑀) = 𝑁)
uptr.j	⊢ 𝐽 = (Hom ‘𝐷)
uptr.m	⊢ (𝜑 → 𝑀 ∈ (𝑋𝐽(𝐹‘𝑍)))

Assertion

Ref	Expression
uptr	⊢ (𝜑 → (𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁))

Proof of Theorem uptr

Step	Hyp	Ref	Expression
1		simpr 490	. 2 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀)
2		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁) → 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁)
3		uptr.y	. . . . 5 ⊢ (𝜑 → (𝑅‘𝑋) = 𝑌)
4	3	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁) → (𝑅‘𝑋) = 𝑌)
5		uptr.r	. . . . 5 ⊢ (𝜑 → 𝑅((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))𝑆)
6	5	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁) → 𝑅((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))𝑆)
7		uptr.k	. . . . 5 ⊢ (𝜑 → (⟨𝑅, 𝑆⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝐾, 𝐿⟩)
8	7	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁) → (⟨𝑅, 𝑆⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝐾, 𝐿⟩)
9		uptr.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
10		uptr.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11	10	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁) → 𝑋 ∈ 𝐵)
12		uptr.f	. . . . 5 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
13	12	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁) → 𝐹(𝐶 Func 𝐷)𝐺)
14		uptr.n	. . . . 5 ⊢ (𝜑 → ((𝑋𝑆(𝐹‘𝑍))‘𝑀) = 𝑁)
15	14	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁) → ((𝑋𝑆(𝐹‘𝑍))‘𝑀) = 𝑁)
16		uptr.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐷)
17		uptr.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐽(𝐹‘𝑍)))
18	17	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁) → 𝑀 ∈ (𝑋𝐽(𝐹‘𝑍)))
19		eqid 2761	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
20	2, 19	uprcl4 50243	. . . 4 ⊢ ((𝜑 ∧ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁) → 𝑍 ∈ (Base‘𝐶))
21	4, 6, 8, 9, 11, 13, 15, 16, 18, 19, 20	uptrlem3 50264	. . 3 ⊢ ((𝜑 ∧ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁) → (𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁))
22	2, 21	mpbird 260	. 2 ⊢ ((𝜑 ∧ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁) → 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀)
23	3	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → (𝑅‘𝑋) = 𝑌)
24	5	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → 𝑅((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))𝑆)
25	7	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → (⟨𝑅, 𝑆⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝐾, 𝐿⟩)
26	10	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → 𝑋 ∈ 𝐵)
27	12	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → 𝐹(𝐶 Func 𝐷)𝐺)
28	14	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → ((𝑋𝑆(𝐹‘𝑍))‘𝑀) = 𝑁)
29	17	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → 𝑀 ∈ (𝑋𝐽(𝐹‘𝑍)))
30	1, 19	uprcl4 50243	. . 3 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → 𝑍 ∈ (Base‘𝐶))
31	23, 24, 25, 9, 26, 27, 28, 16, 29, 19, 30	uptrlem3 50264	. 2 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → (𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁))
32	1, 22, 31	bibiad 853	1 ⊢ (𝜑 → (𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∩ cin 3898  ⟨cop 4590   class class class wbr 5103  ‘cfv 6531  (class class class)co 7412  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:  uptri  50266  uptra  50267  lmddu  50719