Theorem uptrlem3 50031

Description: Lemma for uptr 50032. (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)))
uptrlem3.a	⊢ 𝐴 = (Base‘𝐶)
uptrlem3.z	⊢ (𝜑 → 𝑍 ∈ 𝐴)

Assertion

Ref	Expression
uptrlem3	⊢ (𝜑 → (𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁))

Proof of Theorem uptrlem3

Dummy variables 𝑔 ℎ 𝑘 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2766	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
2		uptr.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐷)
3		eqid 2766	. . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
4		eqid 2766	. . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷)
5		eqid 2766	. . . 4 ⊢ (comp‘𝐸) = (comp‘𝐸)
6		uptr.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7		uptr.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐷)
8	6, 7	eleqtrdi 2876	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))
9	8	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑋 ∈ (Base‘𝐷))
10		uptr.y	. . . . 5 ⊢ (𝜑 → (𝑅‘𝑋) = 𝑌)
11	10	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑅‘𝑋) = 𝑌)
12		uptrlem3.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝐴)
13		uptrlem3.a	. . . . . 6 ⊢ 𝐴 = (Base‘𝐶)
14	12, 13	eleqtrdi 2876	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
15	14	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑍 ∈ (Base‘𝐶))
16		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
17	16, 13	eleqtrdi 2876	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ (Base‘𝐶))
18		uptr.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐽(𝐹‘𝑍)))
19	18	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑀 ∈ (𝑋𝐽(𝐹‘𝑍)))
20		uptr.n	. . . . 5 ⊢ (𝜑 → ((𝑋𝑆(𝐹‘𝑍))‘𝑀) = 𝑁)
21	20	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑋𝑆(𝐹‘𝑍))‘𝑀) = 𝑁)
22		uptr.f	. . . . 5 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
23	22	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐹(𝐶 Func 𝐷)𝐺)
24		uptr.r	. . . . 5 ⊢ (𝜑 → 𝑅((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))𝑆)
25	24	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑅((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))𝑆)
26		uptr.k	. . . . 5 ⊢ (𝜑 → (⟨𝑅, 𝑆⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝐾, 𝐿⟩)
27	26	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (⟨𝑅, 𝑆⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝐾, 𝐿⟩)
28	1, 2, 3, 4, 5, 9, 11, 15, 17, 19, 21, 23, 25, 27	uptrlem1 50029	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (∀ℎ ∈ (𝑌(Hom ‘𝐸)(𝐾‘𝑦))∃!𝑘 ∈ (𝑍(Hom ‘𝐶)𝑦)ℎ = (((𝑍𝐿𝑦)‘𝑘)(⟨𝑌, (𝐾‘𝑍)⟩(comp‘𝐸)(𝐾‘𝑦))𝑁) ↔ ∀𝑔 ∈ (𝑋𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑍(Hom ‘𝐶)𝑦)𝑔 = (((𝑍𝐺𝑦)‘𝑘)(⟨𝑋, (𝐹‘𝑍)⟩(comp‘𝐷)(𝐹‘𝑦))𝑀)))
29	28	ralbidva 3189	. 2 ⊢ (𝜑 → (∀𝑦 ∈ 𝐴 ∀ℎ ∈ (𝑌(Hom ‘𝐸)(𝐾‘𝑦))∃!𝑘 ∈ (𝑍(Hom ‘𝐶)𝑦)ℎ = (((𝑍𝐿𝑦)‘𝑘)(⟨𝑌, (𝐾‘𝑍)⟩(comp‘𝐸)(𝐾‘𝑦))𝑁) ↔ ∀𝑦 ∈ 𝐴 ∀𝑔 ∈ (𝑋𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑍(Hom ‘𝐶)𝑦)𝑔 = (((𝑍𝐺𝑦)‘𝑘)(⟨𝑋, (𝐹‘𝑍)⟩(comp‘𝐷)(𝐹‘𝑦))𝑀)))
30		eqid 2766	. . 3 ⊢ (Base‘𝐸) = (Base‘𝐸)
31		inss1 4192	. . . . . . . . 9 ⊢ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ⊆ (𝐷 Full 𝐸)
32		fullfunc 17990	. . . . . . . . 9 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
33	31, 32	sstri 3949	. . . . . . . 8 ⊢ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ⊆ (𝐷 Func 𝐸)
34	33	ssbri 5161	. . . . . . 7 ⊢ (𝑅((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))𝑆 → 𝑅(𝐷 Func 𝐸)𝑆)
35	24, 34	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑅(𝐷 Func 𝐸)𝑆)
36	7, 30, 35	funcf1 17948	. . . . 5 ⊢ (𝜑 → 𝑅:𝐵⟶(Base‘𝐸))
37	36, 6	ffvelcdmd 7087	. . . 4 ⊢ (𝜑 → (𝑅‘𝑋) ∈ (Base‘𝐸))
38	10, 37	eqeltrrd 2867	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐸))
39	22, 35	cofucla 49915	. . . . 5 ⊢ (𝜑 → (⟨𝑅, 𝑆⟩ ∘func ⟨𝐹, 𝐺⟩) ∈ (𝐶 Func 𝐸))
40	26, 39	eqeltrrd 2867	. . . 4 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐸))
41		df-br 5115	. . . 4 ⊢ (𝐾(𝐶 Func 𝐸)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐸))
42	40, 41	sylibr 237	. . 3 ⊢ (𝜑 → 𝐾(𝐶 Func 𝐸)𝐿)
43	13, 7, 22	funcf1 17948	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
44	43, 12	ffvelcdmd 7087	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑍) ∈ 𝐵)
45	7, 2, 3, 35, 6, 44	funcf2 17950	. . . . 5 ⊢ (𝜑 → (𝑋𝑆(𝐹‘𝑍)):(𝑋𝐽(𝐹‘𝑍))⟶((𝑅‘𝑋)(Hom ‘𝐸)(𝑅‘(𝐹‘𝑍))))
46	45, 18	ffvelcdmd 7087	. . . 4 ⊢ (𝜑 → ((𝑋𝑆(𝐹‘𝑍))‘𝑀) ∈ ((𝑅‘𝑋)(Hom ‘𝐸)(𝑅‘(𝐹‘𝑍))))
47	13, 22, 35, 26, 12	cofu1a 49913	. . . . 5 ⊢ (𝜑 → (𝑅‘(𝐹‘𝑍)) = (𝐾‘𝑍))
48	10, 47	oveq12d 7441	. . . 4 ⊢ (𝜑 → ((𝑅‘𝑋)(Hom ‘𝐸)(𝑅‘(𝐹‘𝑍))) = (𝑌(Hom ‘𝐸)(𝐾‘𝑍)))
49	46, 20, 48	3eltr3d 2880	. . 3 ⊢ (𝜑 → 𝑁 ∈ (𝑌(Hom ‘𝐸)(𝐾‘𝑍)))
50	13, 30, 1, 3, 5, 38, 42, 12, 49	isup 49999	. 2 ⊢ (𝜑 → (𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁 ↔ ∀𝑦 ∈ 𝐴 ∀ℎ ∈ (𝑌(Hom ‘𝐸)(𝐾‘𝑦))∃!𝑘 ∈ (𝑍(Hom ‘𝐶)𝑦)ℎ = (((𝑍𝐿𝑦)‘𝑘)(⟨𝑌, (𝐾‘𝑍)⟩(comp‘𝐸)(𝐾‘𝑦))𝑁)))
51	13, 7, 1, 2, 4, 6, 22, 12, 18	isup 49999	. 2 ⊢ (𝜑 → (𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀 ↔ ∀𝑦 ∈ 𝐴 ∀𝑔 ∈ (𝑋𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑍(Hom ‘𝐶)𝑦)𝑔 = (((𝑍𝐺𝑦)‘𝑘)(⟨𝑋, (𝐹‘𝑍)⟩(comp‘𝐷)(𝐹‘𝑦))𝑀)))
52	29, 50, 51	3bitr4rd 315	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  ∀wral 3082  ∃!wreu 3370   ∩ cin 3907  ⟨cop 4600   class class class wbr 5114  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  Hom chom 17346  compcco 17347   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:  uptr  50032