Theorem uptri 49455

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⟩)
uptri.n	⊢ (𝜑 → ((𝑋𝑆(𝐹‘𝑍))‘𝑀) = 𝑁)
uptri.z	⊢ (𝜑 → 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀)

Assertion

Ref	Expression
uptri	⊢ (𝜑 → 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁)

Proof of Theorem uptri

Step	Hyp	Ref	Expression
1		uptri.z	. 2 ⊢ (𝜑 → 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀)
2		uptr.y	. . . . 5 ⊢ (𝜑 → (𝑅‘𝑋) = 𝑌)
3	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → (𝑅‘𝑋) = 𝑌)
4		uptr.r	. . . . 5 ⊢ (𝜑 → 𝑅((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))𝑆)
5	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → 𝑅((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))𝑆)
6		uptr.k	. . . . 5 ⊢ (𝜑 → (⟨𝑅, 𝑆⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝐾, 𝐿⟩)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → (⟨𝑅, 𝑆⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝐾, 𝐿⟩)
8		eqid 2736	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
9	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀)
10	9, 8	uprcl3 49431	. . . 4 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → 𝑋 ∈ (Base‘𝐷))
11	9	uprcl2 49430	. . . 4 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → 𝐹(𝐶 Func 𝐷)𝐺)
12		uptri.n	. . . . 5 ⊢ (𝜑 → ((𝑋𝑆(𝐹‘𝑍))‘𝑀) = 𝑁)
13	12	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → ((𝑋𝑆(𝐹‘𝑍))‘𝑀) = 𝑁)
14		eqid 2736	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
15	9, 14	uprcl5 49433	. . . 4 ⊢ ((𝜑 ∧ 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) → 𝑀 ∈ (𝑋(Hom ‘𝐷)(𝐹‘𝑍)))
16	3, 5, 7, 8, 10, 11, 13, 14, 15	uptr 49454	. . 3 ⊢ ((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))
17	1, 16	mpdan 687	. 2 ⊢ (𝜑 → (𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁))
18	1, 17	mpbid 232	1 ⊢ (𝜑 → 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∩ cin 3900  ⟨cop 4586   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  Hom chom 17188   ∘_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: (None)