Theorem upeu2lem 48884

Description: Lemma for upeu2 48902. 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 17816	. . . 4 ⊢ (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌𝐻𝑋))
10		eqid 2736	. . . . . 6 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
11	1, 10, 4, 6, 5, 8	invf 17808	. . . . 5 ⊢ (𝜑 → (𝑋(Inv‘𝐶)𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
12		upeu2lem.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))
13	11, 12	ffvelcdmd 7103	. . . 4 ⊢ (𝜑 → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌𝐼𝑋))
14	9, 13	sseldd 3983	. . 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 17724	. 2 ⊢ (𝜑 → (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) ∈ (𝑌𝐻𝑍))
17		oveq1 7436	. . . . . 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 481	. . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → 𝐶 ∈ Cat)
20	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → 𝑌 ∈ 𝐵)
21	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → 𝑋 ∈ 𝐵)
22	14	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌𝐻𝑋))
23	1, 2, 8, 4, 6, 5	isohom 17816	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋𝐻𝑌))
24	23, 12	sseldd 3983	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
25	24	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → 𝐹 ∈ (𝑋𝐻𝑌))
26	7	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → 𝑍 ∈ 𝐵)
27		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → 𝑘 ∈ (𝑌𝐻𝑍))
28	1, 2, 3, 19, 20, 21, 20, 22, 25, 26, 27	catass 17725	. . . . . . 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 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → 𝐹 ∈ (𝑋𝐼𝑌))
30		eqid 2736	. . . . . . . . 9 ⊢ (Id‘𝐶) = (Id‘𝐶)
31	3	oveqi 7442	. . . . . . . . 9 ⊢ (⟨𝑌, 𝑋⟩ · 𝑌) = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
32	1, 8, 10, 19, 21, 20, 29, 30, 31	isocoinvid 17833	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = ((Id‘𝐶)‘𝑌))
33	32	oveq2d 7445	. . . . . . 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 17723	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → (𝑘(⟨𝑌, 𝑌⟩ · 𝑍)((Id‘𝐶)‘𝑌)) = 𝑘)
35	28, 33, 34	3eqtrd 2780	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → ((𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = 𝑘)
36	35	adantr 480	. . . . 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 2777	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) ∧ 𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)) → 𝑘 = (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
38		oveq1 7436	. . . . . 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 481	. . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → 𝐺 ∈ (𝑋𝐻𝑍))
41	1, 2, 3, 19, 21, 20, 21, 25, 22, 26, 40	catass 17725	. . . . . . 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 7442	. . . . . . . . 9 ⊢ (⟨𝑋, 𝑌⟩ · 𝑋) = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)
43	1, 8, 10, 19, 21, 20, 29, 30, 42	invcoisoid 17832	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹)(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
44	43	oveq2d 7445	. . . . . . 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 17723	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → (𝐺(⟨𝑋, 𝑋⟩ · 𝑍)((Id‘𝐶)‘𝑋)) = 𝐺)
46	41, 44, 45	3eqtrd 2780	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → ((𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹))(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = 𝐺)
47	46	adantr 480	. . . . 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 2777	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) ∧ 𝑘 = (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹))) → 𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))
49	37, 48	impbida 801	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → (𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ↔ 𝑘 = (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹))))
50	49	ralrimiva 3145	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (𝑌𝐻𝑍)(𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ↔ 𝑘 = (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹))))
51		reu6i 3733	. 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 584	1 ⊢ (𝜑 → ∃!𝑘 ∈ (𝑌𝐻𝑍)𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3060  ∃!wreu 3377  ⟨cop 4630  ‘cfv 6559  (class class class)co 7429  Basecbs 17243  Hom chom 17304  compcco 17305  Catccat 17703  Idccid 17704  Invcinv 17785  Isociso 17786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-riota 7386  df-ov 7432  df-oprab 7433  df-mpo 7434  df-1st 8010  df-2nd 8011  df-cat 17707  df-cid 17708  df-sect 17787  df-inv 17788  df-iso 17789

This theorem is referenced by:  upeu2  48902