Theorem uptr 49454

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 484	. 2 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀)
2		simpr 484	. . 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 480	. . . 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 480	. . . 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 480	. . . 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 480	. . . 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 480	. . . 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 480	. . . 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 480	. . . 4 ⊢ ((𝜑 ∧ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁) → 𝑀 ∈ (𝑋𝐽(𝐹‘𝑍)))
19		eqid 2736	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
20	2, 19	uprcl4 49432	. . . 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 49453	. . 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 257	. 2 ⊢ ((𝜑 ∧ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁) → 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀)
23	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → (𝑅‘𝑋) = 𝑌)
24	5	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → 𝑅((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))𝑆)
25	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → (⟨𝑅, 𝑆⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝐾, 𝐿⟩)
26	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → 𝑋 ∈ 𝐵)
27	12	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → 𝐹(𝐶 Func 𝐷)𝐺)
28	14	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → ((𝑋𝑆(𝐹‘𝑍))‘𝑀) = 𝑁)
29	17	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → 𝑀 ∈ (𝑋𝐽(𝐹‘𝑍)))
30	1, 19	uprcl4 49432	. . 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 49453	. 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 839	1 ⊢ (𝜑 → (𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ∩ cin 3900  ⟨cop 4586   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  Hom chom 17188   Func cfunc 17778   ∘_func ccofu 17780   Full cful 17828   Faith cfth 17829   UP cup 49414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8765  df-ixp 8836  df-cat 17591  df-cid 17592  df-func 17782  df-cofu 17784  df-full 17830  df-fth 17831  df-up 49415

This theorem is referenced by:  uptri  49455  uptra  49456  lmddu  49908