Theorem upeu2lem 49532

Description: Lemma for upeu2 49676. 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 17738	. . . 4 ⊢ (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌𝐻𝑋))
10		eqid 2741	. . . . . 6 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
11	1, 10, 4, 6, 5, 8	invf 17730	. . . . 5 ⊢ (𝜑 → (𝑋(Inv‘𝐶)𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
12		upeu2lem.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))
13	11, 12	ffvelcdmd 7030	. . . 4 ⊢ (𝜑 → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌𝐼𝑋))
14	9, 13	sseldd 3918	. . 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 17646	. 2 ⊢ (𝜑 → (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) ∈ (𝑌𝐻𝑍))
17		oveq1 7367	. . . . . 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 483	. . . . 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 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → 𝐶 ∈ Cat)
20	5	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → 𝑌 ∈ 𝐵)
21	6	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → 𝑋 ∈ 𝐵)
22	14	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌𝐻𝑋))
23	1, 2, 8, 4, 6, 5	isohom 17738	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋𝐻𝑌))
24	23, 12	sseldd 3918	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
25	24	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → 𝐹 ∈ (𝑋𝐻𝑌))
26	7	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → 𝑍 ∈ 𝐵)
27		simpr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → 𝑘 ∈ (𝑌𝐻𝑍))
28	1, 2, 3, 19, 20, 21, 20, 22, 25, 26, 27	catass 17647	. . . . . . 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 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → 𝐹 ∈ (𝑋𝐼𝑌))
30		eqid 2741	. . . . . . . . 9 ⊢ (Id‘𝐶) = (Id‘𝐶)
31	3	oveqi 7373	. . . . . . . . 9 ⊢ (⟨𝑌, 𝑋⟩ · 𝑌) = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
32	1, 8, 10, 19, 21, 20, 29, 30, 31	isocoinvid 17755	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = ((Id‘𝐶)‘𝑌))
33	32	oveq2d 7376	. . . . . . 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 17645	. . . . . . 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 482	. . . . 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 7367	. . . . . 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 483	. . . . 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 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → 𝐺 ∈ (𝑋𝐻𝑍))
41	1, 2, 3, 19, 21, 20, 21, 25, 22, 26, 40	catass 17647	. . . . . . 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 7373	. . . . . . . . 9 ⊢ (⟨𝑋, 𝑌⟩ · 𝑋) = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)
43	1, 8, 10, 19, 21, 20, 29, 30, 42	invcoisoid 17754	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹)(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
44	43	oveq2d 7376	. . . . . . 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 17645	. . . . . . 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 482	. . . . 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 807	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑌𝐻𝑍)) → (𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ↔ 𝑘 = (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹))))
50	49	ralrimiva 3133	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (𝑌𝐻𝑍)(𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ↔ 𝑘 = (𝐺(⟨𝑌, 𝑋⟩ · 𝑍)((𝑋(Inv‘𝐶)𝑌)‘𝐹))))
51		reu6i 3671	. 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 591	1 ⊢ (𝜑 → ∃!𝑘 ∈ (𝑌𝐻𝑍)𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∀wral 3055  ∃!wreu 3344  ⟨cop 4564  ‘cfv 6489  (class class class)co 7360  Basecbs 17174  Hom chom 17226  compcco 17227  Catccat 17625  Idccid 17626  Invcinv 17707  Isociso 17708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-cat 17629  df-cid 17630  df-sect 17709  df-inv 17710  df-iso 17711

This theorem is referenced by:  upeu2  49676