Theorem upeu2lem 49954

Description: Lemma for upeu2 50098. There exists a unique morphism from 𝑌 to 𝑍 that commutes if 𝐹:𝑋⟶𝑌 is an isomorphism. (Contributed by Zhi Wang, 20-Sep-2025.)

Hypotheses

Ref	Expression
upeu2lem.b	⊢ 𝐵 = (Base‘𝐶)
upeu2lem.h	⊢ 𝐻 = (Hom ‘𝐶)
upeu2lem.o	⊢ · = (comp‘𝐶)
upeu2lem.i	⊢ 𝐼 = (Iso‘𝐶)
upeu2lem.c	⊢ (𝜑 → 𝐶 ∈ Cat)
upeu2lem.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
upeu2lem.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
upeu2lem.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
upeu2lem.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))
upeu2lem.g	⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑍))

Assertion

Ref	Expression
upeu2lem	⊢ (𝜑 → ∃!𝑘 ∈ (𝑌𝐻𝑍)𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))

Distinct variable groups:   · ,𝑘   𝐶,𝑘   𝑘,𝐹   𝑘,𝐺   𝑘,𝐻   𝑘,𝑋   𝑘,𝑌   𝑘,𝑍   𝜑,𝑘

Allowed substitution hints:   𝐵(𝑘)   𝐼(𝑘)

Proof of Theorem upeu2lem

Step	Hyp	Ref	Expression
1		upeu2lem.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		upeu2lem.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
3		upeu2lem.o	. . 3 ⊢ · = (comp‘𝐶)
4		upeu2lem.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		upeu2lem.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6		upeu2lem.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7		upeu2lem.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
8		upeu2lem.i	. . . . 5 ⊢ 𝐼 = (Iso‘𝐶)
9	1, 2, 8, 4, 5, 6	isohom 17865	. . . 4 ⊢ (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌𝐻𝑋))
10		eqid 2760	. . . . . 6 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
11	1, 10, 4, 6, 5, 8	invf 17857	. . . . 5 ⊢ (𝜑 → (𝑋(Inv‘𝐶)𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
12		upeu2lem.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))
13	11, 12	ffvelcdmd 7078	. . . 4 ⊢ (𝜑 → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌𝐼𝑋))
14	9, 13	sseldd 3932	. . 3 ⊢ (𝜑 → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌𝐻𝑋))
15		upeu2lem.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑍))
16	1, 2, 3, 4, 5, 6, 7, 14, 15	catcocl 17773	. 2 ⊢ (𝜑 → (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) ∈ (𝑌𝐻𝑍))
17		oveq1 7420	. . . . . 6 ⊢ (𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) → (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = ((𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
18	17	adantl 487	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) ∧ 𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)) → (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = ((𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
19	4	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → 𝐶 ∈ Cat)
20	5	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → 𝑌 ∈ 𝐵)
21	6	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → 𝑋 ∈ 𝐵)
22	14	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌𝐻𝑋))
23	1, 2, 8, 4, 6, 5	isohom 17865	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋𝐻𝑌))
24	23, 12	sseldd 3932	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
25	24	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → 𝐹 ∈ (𝑋𝐻𝑌))
26	7	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → 𝑍 ∈ 𝐵)
27		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → 𝑘 ∈ (𝑌𝐻𝑍))
28	1, 2, 3, 19, 20, 21, 20, 22, 25, 26, 27	catass 17774	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → ((𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = (𝑘(⟨𝑌, 𝑌⟩ · 𝑍)(𝐹(⟨𝑌, 𝑋⟩ · 𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹))))
29	12	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → 𝐹 ∈ (𝑋𝐼𝑌))
30		eqid 2760	. . . . . . . . 9 ⊢ (Id‘𝐶) = (Id‘𝐶)
31	3	oveqi 7426	. . . . . . . . 9 ⊢ (⟨𝑌, 𝑋⟩ · 𝑌) = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
32	1, 8, 10, 19, 21, 20, 29, 30, 31	isocoinvid 17882	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = ((Id‘𝐶)‘𝑌))
33	32	oveq2d 7429	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → (𝑘(⟨𝑌, 𝑌⟩ · 𝑍)(𝐹(⟨𝑌, 𝑋⟩ · 𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = (𝑘(⟨𝑌, 𝑌⟩ · 𝑍)((Id‘𝐶)‘𝑌)))
34	1, 2, 30, 19, 20, 3, 26, 27	catrid 17772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → (𝑘(⟨𝑌, 𝑌⟩ · 𝑍)((Id‘𝐶)‘𝑌)) = 𝑘)
35	28, 33, 34	3eqtrd 2799	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → ((𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = 𝑘)
36	35	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) ∧ 𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)) → ((𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = 𝑘)
37	18, 36	eqtr2d 2796	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) ∧ 𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)) → 𝑘 = (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
38		oveq1 7420	. . . . . 6 ⊢ (𝑘 = (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) → (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = ((𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹))(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))
39	38	adantl 487	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) ∧ 𝑘 = (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹))) → (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = ((𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹))(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))
40	15	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → 𝐺 ∈ (𝑋𝐻𝑍))
41	1, 2, 3, 19, 21, 20, 21, 25, 22, 26, 40	catass 17774	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → ((𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹))(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺(⟨𝑋, 𝑋⟩ · 𝑍)(((𝑋(Inv‘𝐶)𝑌)‘𝐹)(⟨𝑋, 𝑌⟩ · 𝑋)𝐹)))
42	3	oveqi 7426	. . . . . . . . 9 ⊢ (⟨𝑋, 𝑌⟩ · 𝑋) = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)
43	1, 8, 10, 19, 21, 20, 29, 30, 42	invcoisoid 17881	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹)(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
44	43	oveq2d 7429	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → (𝐺(⟨𝑋, 𝑋⟩ · 𝑍)(((𝑋(Inv‘𝐶)𝑌)‘𝐹)(⟨𝑋, 𝑌⟩ · 𝑋)𝐹)) = (𝐺(⟨𝑋, 𝑋⟩ · 𝑍)((Id‘𝐶)‘𝑋)))
45	1, 2, 30, 19, 21, 3, 26, 40	catrid 17772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → (𝐺(⟨𝑋, 𝑋⟩ · 𝑍)((Id‘𝐶)‘𝑋)) = 𝐺)
46	41, 44, 45	3eqtrd 2799	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → ((𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹))(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = 𝐺)
47	46	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) ∧ 𝑘 = (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹))) → ((𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹))(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = 𝐺)
48	39, 47	eqtr2d 2796	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) ∧ 𝑘 = (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹))) → 𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))
49	37, 48	impbida 813	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → (𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ↔ 𝑘 = (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹))))
50	49	ralrimiva 3154	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (𝑌𝐻𝑍)(𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ↔ 𝑘 = (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹))))
51		reu6i 3686	. 2 ⊢ (((𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) ∈ (𝑌𝐻𝑍) ∧ ∀𝑘 ∈ (𝑌𝐻𝑍)(𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ↔ 𝑘 = (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹)))) → ∃!𝑘 ∈ (𝑌𝐻𝑍)𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))
52	16, 50, 51	syl2anc 596	1 ⊢ (𝜑 → ∃!𝑘 ∈ (𝑌𝐻𝑍)𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃!wreu 3363  ⟨cop 4590  ‘cfv 6533  (class class class)co 7413  Basecbs 17301  Hom chom 17353  compcco 17354  Catccat 17752  Idccid 17753  Invcinv 17834  Isociso 17835

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-cat 17756  df-cid 17757  df-sect 17836  df-inv 17837  df-iso 17838

This theorem is used by:  upeu2  50098