Theorem upeu2lem 49013

Description: Lemma for upeu2 49157. 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 17683	. . . 4 ⊢ (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌𝐻𝑋))
10		eqid 2729	. . . . . 6 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
11	1, 10, 4, 6, 5, 8	invf 17675	. . . . 5 ⊢ (𝜑 → (𝑋(Inv‘𝐶)𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
12		upeu2lem.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))
13	11, 12	ffvelcdmd 7019	. . . 4 ⊢ (𝜑 → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌𝐼𝑋))
14	9, 13	sseldd 3936	. . 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 17591	. 2 ⊢ (𝜑 → (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) ∈ (𝑌𝐻𝑍))
17		oveq1 7356	. . . . . 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 17683	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋𝐻𝑌))
24	23, 12	sseldd 3936	. . . . . . . . 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 17592	. . . . . . 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 2729	. . . . . . . . 9 ⊢ (Id‘𝐶) = (Id‘𝐶)
31	3	oveqi 7362	. . . . . . . . 9 ⊢ (⟨𝑌, 𝑋⟩ · 𝑌) = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
32	1, 8, 10, 19, 21, 20, 29, 30, 31	isocoinvid 17700	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = ((Id‘𝐶)‘𝑌))
33	32	oveq2d 7365	. . . . . . 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 17590	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → (𝑘(⟨𝑌, 𝑌⟩ · 𝑍)((Id‘𝐶)‘𝑌)) = 𝑘)
35	28, 33, 34	3eqtrd 2768	. . . . . 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 2765	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) ∧ 𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)) → 𝑘 = (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
38		oveq1 7356	. . . . . 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 17592	. . . . . . 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 7362	. . . . . . . . 9 ⊢ (⟨𝑋, 𝑌⟩ · 𝑋) = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)
43	1, 8, 10, 19, 21, 20, 29, 30, 42	invcoisoid 17699	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹)(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
44	43	oveq2d 7365	. . . . . . 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 17590	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → (𝐺(⟨𝑋, 𝑋⟩ · 𝑍)((Id‘𝐶)‘𝑋)) = 𝐺)
46	41, 44, 45	3eqtrd 2768	. . . . . 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 2765	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) ∧ 𝑘 = (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹))) → 𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))
49	37, 48	impbida 800	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → (𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ↔ 𝑘 = (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹))))
50	49	ralrimiva 3121	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (𝑌𝐻𝑍)(𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ↔ 𝑘 = (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹))))
51		reu6i 3688	. 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 2109  ∀wral 3044  ∃!wreu 3341  ⟨cop 4583  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  Hom chom 17172  compcco 17173  Catccat 17570  Idccid 17571  Invcinv 17652  Isociso 17653

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-cat 17574  df-cid 17575  df-sect 17654  df-inv 17655  df-iso 17656

This theorem is referenced by:  upeu2  49157