Theorem uptr 49688

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 49666	. . . 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 49687	. . 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 49666	. . 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 49687	. 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 840	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 1542   ∈ wcel 2114   ∩ cin 3888  ⟨cop 4573   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  Hom chom 17231   Func cfunc 17821   ∘_func ccofu 17823   Full cful 17871   Faith cfth 17872   UP cup 49648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-map 8775  df-ixp 8846  df-cat 17634  df-cid 17635  df-func 17825  df-cofu 17827  df-full 17873  df-fth 17874  df-up 49649

This theorem is referenced by:  uptri  49689  uptra  49690  lmddu  50142