Theorem upciclem1 48776

Description: Lemma for upcic 48780, upeu 48781, and upeu2 48782. (Contributed by Zhi Wang, 16-Sep-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 2744	. . . 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 3406	. . 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 6924	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
4	3	oveq2d 7468	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑍𝐽(𝐹‘𝑦)) = (𝑍𝐽(𝐹‘𝑌)))
5	3	oveq2d 7468	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦)) = (⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌)))
6		oveq2 7460	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → (𝑋𝐺𝑦) = (𝑋𝐺𝑌))
7	6	fveq1d 6926	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → ((𝑋𝐺𝑦)‘𝑘) = ((𝑋𝐺𝑌)‘𝑘))
8		eqidd 2741	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → 𝑀 = 𝑀)
9	5, 7, 8	oveq123d 7473	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (((𝑋𝐺𝑦)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀) = (((𝑋𝐺𝑌)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
10	9	eqeq2d 2751	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑛 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀) ↔ 𝑛 = (((𝑋𝐺𝑌)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)))
11	10	reubidv 3406	. . . . . 6 ⊢ (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)))
12		oveq2 7460	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌))
13	12	reueqdv 3431	. . . . . 6 ⊢ (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)))
14	11, 13	bitrd 279	. . . . 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)))
15	4, 14	raleqbidv 3354	. . . 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)))
16		upciclem1.1	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑛 ∈ (𝑍𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑋𝐻𝑦)𝑛 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀))
17		upciclem1.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
18	15, 16, 17	rspcdva 3637	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ (𝑍𝐽(𝐹‘𝑌))∃!𝑘 ∈ (𝑋𝐻𝑌)𝑛 = (((𝑋𝐺𝑌)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
19		upciclem1.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ (𝑍𝐽(𝐹‘𝑌)))
20	2, 18, 19	rspcdva 3637	. 2 ⊢ (𝜑 → ∃!𝑘 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
21		fveq2 6924	. . . . . 6 ⊢ (𝑘 = 𝑚 → ((𝑋𝐺𝑌)‘𝑘) = ((𝑋𝐺𝑌)‘𝑚))
22	21	oveq1d 7467	. . . . 5 ⊢ (𝑘 = 𝑚 → (((𝑋𝐺𝑌)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) = (((𝑋𝐺𝑌)‘𝑚)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
23	22	eqeq2d 2751	. . . 4 ⊢ (𝑘 = 𝑚 → (𝑁 = (((𝑋𝐺𝑌)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) ↔ 𝑁 = (((𝑋𝐺𝑌)‘𝑚)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)))
24	23	cbvreuvw 3412	. . 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))
25		fveq2 6924	. . . . . 6 ⊢ (𝑚 = 𝑙 → ((𝑋𝐺𝑌)‘𝑚) = ((𝑋𝐺𝑌)‘𝑙))
26	25	oveq1d 7467	. . . . 5 ⊢ (𝑚 = 𝑙 → (((𝑋𝐺𝑌)‘𝑚)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) = (((𝑋𝐺𝑌)‘𝑙)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
27	26	eqeq2d 2751	. . . 4 ⊢ (𝑚 = 𝑙 → (𝑁 = (((𝑋𝐺𝑌)‘𝑚)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀) ↔ 𝑁 = (((𝑋𝐺𝑌)‘𝑙)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)))
28	27	cbvreuvw 3412	. . 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))
29	24, 28	bitri 275	. 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))
30	20, 29	sylib 218	1 ⊢ (𝜑 → ∃!𝑙 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑙)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃!wreu 3386  ⟨cop 4655  ‘cfv 6577  (class class class)co 7452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3445  df-v 3491  df-dif 3980  df-un 3982  df-ss 3994  df-nul 4354  df-if 4550  df-sn 4650  df-pr 4652  df-op 4656  df-uni 4934  df-br 5169  df-iota 6529  df-fv 6585  df-ov 7455

This theorem is referenced by:  upciclem3  48778  upciclem4  48779  upeu  48781  upeu2  48782