Theorem upeu2lem 49060

Description: Lemma for upeu2 49204. 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 17678	. . . 4 ⊢ (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌𝐻𝑋))
10		eqid 2731	. . . . . 6 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
11	1, 10, 4, 6, 5, 8	invf 17670	. . . . 5 ⊢ (𝜑 → (𝑋(Inv‘𝐶)𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
12		upeu2lem.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))
13	11, 12	ffvelcdmd 7013	. . . 4 ⊢ (𝜑 → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌𝐼𝑋))
14	9, 13	sseldd 3930	. . 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 17586	. 2 ⊢ (𝜑 → (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) ∈ (𝑌𝐻𝑍))
17		oveq1 7348	. . . . . 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 17678	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋𝐻𝑌))
24	23, 12	sseldd 3930	. . . . . . . . 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 17587	. . . . . . 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 2731	. . . . . . . . 9 ⊢ (Id‘𝐶) = (Id‘𝐶)
31	3	oveqi 7354	. . . . . . . . 9 ⊢ (⟨𝑌, 𝑋⟩ · 𝑌) = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
32	1, 8, 10, 19, 21, 20, 29, 30, 31	isocoinvid 17695	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = ((Id‘𝐶)‘𝑌))
33	32	oveq2d 7357	. . . . . . 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 17585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → (𝑘(⟨𝑌, 𝑌⟩ · 𝑍)((Id‘𝐶)‘𝑌)) = 𝑘)
35	28, 33, 34	3eqtrd 2770	. . . . . 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 2767	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) ∧ 𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)) → 𝑘 = (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
38		oveq1 7348	. . . . . 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 17587	. . . . . . 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 7354	. . . . . . . . 9 ⊢ (⟨𝑋, 𝑌⟩ · 𝑋) = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)
43	1, 8, 10, 19, 21, 20, 29, 30, 42	invcoisoid 17694	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹)(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
44	43	oveq2d 7357	. . . . . . 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 17585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → (𝐺(⟨𝑋, 𝑋⟩ · 𝑍)((Id‘𝐶)‘𝑋)) = 𝐺)
46	41, 44, 45	3eqtrd 2770	. . . . . 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 2767	. . . 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 3124	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (𝑌𝐻𝑍)(𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ↔ 𝑘 = (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹))))
51		reu6i 3682	. 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 1541   ∈ wcel 2111  ∀wral 3047  ∃!wreu 3344  ⟨cop 4577  ‘cfv 6476  (class class class)co 7341  Basecbs 17115  Hom chom 17167  compcco 17168  Catccat 17565  Idccid 17566  Invcinv 17647  Isociso 17648

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-cat 17569  df-cid 17570  df-sect 17649  df-inv 17650  df-iso 17651

This theorem is referenced by:  upeu2  49204