Theorem uptr 50032

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 2766	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
20	2, 19	uprcl4 50010	. . . 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 50031	. . 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 50010	. . 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 50031	. 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 2146   ∩ cin 3907  ⟨cop 4600   class class class wbr 5114  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  Hom chom 17346   Func cfunc 17936   ∘_func ccofu 17938   Full cful 17986   Faith cfth 17987   UP cup 49992

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 7995  df-2nd 7996  df-map 8835  df-ixp 8905  df-cat 17749  df-cid 17750  df-func 17940  df-cofu 17942  df-full 17988  df-fth 17989  df-up 49993

This theorem is used by:  uptri  50033  uptra  50034  lmddu  50486