Theorem upciclem1 49824

Description: Lemma for upcic 49828, upeu 49829, and upeu2 49830. (Contributed by Zhi Wang, 16-Sep-2025.) (Proof shortened by Zhi Wang, 5-Nov-2025.)

Hypotheses

Ref	Expression
upciclem1.1	⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑛 ∈ (𝑍𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑋𝐻𝑦)𝑛 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀))
upciclem1.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
upciclem1.n	⊢ (𝜑 → 𝑁 ∈ (𝑍𝐽(𝐹‘𝑌)))

Assertion

Ref	Expression
upciclem1	⊢ (𝜑 → ∃!𝑙 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑙)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))

Distinct variable groups:   𝑦,𝐵   𝑘,𝐹   𝐹,𝑙   𝑛,𝐹,𝑦,𝑘   𝑘,𝐺   𝐺,𝑙   𝑛,𝐺,𝑦   𝑘,𝐻   𝐻,𝑙   𝑛,𝐻,𝑦   𝑛,𝐽,𝑦   𝑘,𝑀   𝑀,𝑙   𝑛,𝑀,𝑦   𝑘,𝑁   𝑁,𝑙   𝑛,𝑁   𝑘,𝑂   𝑂,𝑙   𝑛,𝑂,𝑦   𝑘,𝑋   𝑋,𝑙   𝑛,𝑋,𝑦   𝑘,𝑌   𝑌,𝑙   𝑛,𝑌,𝑦   𝑘,𝑍   𝑍,𝑙   𝑛,𝑍,𝑦

Allowed substitution hints:   𝜑(𝑦,𝑘,𝑛,𝑙)   𝐵(𝑘,𝑛,𝑙)   𝐽(𝑘,𝑙)   𝑁(𝑦)

Proof of Theorem upciclem1

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2773	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑛 = (((𝑋𝐺𝑌)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) ↔ 𝑁 = (((𝑋𝐺𝑌)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)))
2	1	reubidv 3392	. . 3 ⊢ (𝑛 = 𝑁 → (∃!𝑘 ∈ (𝑋𝐻𝑌)𝑛 = (((𝑋𝐺𝑌)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) ↔ ∃!𝑘 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)))
3		fveq2 6879	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
4	3	oveq2d 7424	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑍𝐽(𝐹‘𝑦)) = (𝑍𝐽(𝐹‘𝑌)))
5		oveq2 7416	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌))
6	3	oveq2d 7424	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦)) = (⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌)))
7		oveq2 7416	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (𝑋𝐺𝑦) = (𝑋𝐺𝑌))
8	7	fveq1d 6881	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → ((𝑋𝐺𝑦)‘𝑘) = ((𝑋𝐺𝑌)‘𝑘))
9		eqidd 2770	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → 𝑀 = 𝑀)
10	6, 8, 9	oveq123d 7429	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (((𝑋𝐺𝑦)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀) = (((𝑋𝐺𝑌)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
11	10	eqeq2d 2780	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑛 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀) ↔ 𝑛 = (((𝑋𝐺𝑌)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)))
12	5, 11	reueqbidv 3412	. . . . 5 ⊢ (𝑦 = 𝑌 → (∃!𝑘 ∈ (𝑋𝐻𝑦)𝑛 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀) ↔ ∃!𝑘 ∈ (𝑋𝐻𝑌)𝑛 = (((𝑋𝐺𝑌)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)))
13	4, 12	raleqbidv 3345	. . . 4 ⊢ (𝑦 = 𝑌 → (∀𝑛 ∈ (𝑍𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑋𝐻𝑦)𝑛 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀) ↔ ∀𝑛 ∈ (𝑍𝐽(𝐹‘𝑌))∃!𝑘 ∈ (𝑋𝐻𝑌)𝑛 = (((𝑋𝐺𝑌)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)))
14		upciclem1.1	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑛 ∈ (𝑍𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑋𝐻𝑦)𝑛 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀))
15		upciclem1.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
16	13, 14, 15	rspcdva 3591	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ (𝑍𝐽(𝐹‘𝑌))∃!𝑘 ∈ (𝑋𝐻𝑌)𝑛 = (((𝑋𝐺𝑌)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
17		upciclem1.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ (𝑍𝐽(𝐹‘𝑌)))
18	2, 16, 17	rspcdva 3591	. 2 ⊢ (𝜑 → ∃!𝑘 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
19		fveq2 6879	. . . . . 6 ⊢ (𝑘 = 𝑚 → ((𝑋𝐺𝑌)‘𝑘) = ((𝑋𝐺𝑌)‘𝑚))
20	19	oveq1d 7423	. . . . 5 ⊢ (𝑘 = 𝑚 → (((𝑋𝐺𝑌)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) = (((𝑋𝐺𝑌)‘𝑚)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
21	20	eqeq2d 2780	. . . 4 ⊢ (𝑘 = 𝑚 → (𝑁 = (((𝑋𝐺𝑌)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) ↔ 𝑁 = (((𝑋𝐺𝑌)‘𝑚)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)))
22	21	cbvreuvw 3398	. . 3 ⊢ (∃!𝑘 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) ↔ ∃!𝑚 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑚)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
23		fveq2 6879	. . . . . 6 ⊢ (𝑚 = 𝑙 → ((𝑋𝐺𝑌)‘𝑚) = ((𝑋𝐺𝑌)‘𝑙))
24	23	oveq1d 7423	. . . . 5 ⊢ (𝑚 = 𝑙 → (((𝑋𝐺𝑌)‘𝑚)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) = (((𝑋𝐺𝑌)‘𝑙)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
25	24	eqeq2d 2780	. . . 4 ⊢ (𝑚 = 𝑙 → (𝑁 = (((𝑋𝐺𝑌)‘𝑚)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) ↔ 𝑁 = (((𝑋𝐺𝑌)‘𝑙)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)))
26	25	cbvreuvw 3398	. . 3 ⊢ (∃!𝑚 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑚)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) ↔ ∃!𝑙 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑙)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
27	22, 26	bitri 278	. 2 ⊢ (∃!𝑘 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) ↔ ∃!𝑙 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑙)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
28	18, 27	sylib 221	1 ⊢ (𝜑 → ∃!𝑙 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑙)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃!wreu 3374  ⟨cop 4597  ‘cfv 6534  (class class class)co 7408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-reu 3377  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-iota 6490  df-fv 6542  df-ov 7411

This theorem is referenced by:  upciclem3  49826  upciclem4  49827  upeu  49829  upeu2  49830