Theorem oppcup3lem 49002

Description: Lemma for oppcup3 49005. (Contributed by Zhi Wang, 4-Nov-2025.)

Hypotheses

Ref	Expression
oppcup3lem.1	⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑛 ∈ ((𝐹‘𝑦)𝐽𝑍)∃!𝑘 ∈ (𝑦𝐻𝑋)𝑛 = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩𝑂𝑍)((𝑦𝐺𝑋)‘𝑘)))
oppcup3lem.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
oppcup3lem.n	⊢ (𝜑 → 𝑁 ∈ ((𝐹‘𝑌)𝐽𝑍))

Assertion

Ref	Expression
oppcup3lem	⊢ (𝜑 → ∃!𝑙 ∈ (𝑌𝐻𝑋)𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑙)))

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 oppcup3lem

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2738	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑛 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑘)) ↔ 𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑘))))
2	1	reubidv 3375	. . 3 ⊢ (𝑛 = 𝑁 → (∃!𝑘 ∈ (𝑌𝐻𝑋)𝑛 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑘)) ↔ ∃!𝑘 ∈ (𝑌𝐻𝑋)𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑘))))
3		fveq2 6873	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
4	3	oveq1d 7415	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑦)𝐽𝑍) = ((𝐹‘𝑌)𝐽𝑍))
5		oveq1 7407	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑦𝐻𝑋) = (𝑌𝐻𝑋))
6	3	opeq1d 4853	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → ⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩ = ⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩)
7	6	oveq1d 7415	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩𝑂𝑍) = (⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍))
8		eqidd 2735	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → 𝑀 = 𝑀)
9		oveq1 7407	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (𝑦𝐺𝑋) = (𝑌𝐺𝑋))
10	9	fveq1d 6875	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → ((𝑦𝐺𝑋)‘𝑘) = ((𝑌𝐺𝑋)‘𝑘))
11	7, 8, 10	oveq123d 7421	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩𝑂𝑍)((𝑦𝐺𝑋)‘𝑘)) = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑘)))
12	11	eqeq2d 2745	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑛 = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩𝑂𝑍)((𝑦𝐺𝑋)‘𝑘)) ↔ 𝑛 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑘))))
13	5, 12	reueqbidv 3400	. . . . 5 ⊢ (𝑦 = 𝑌 → (∃!𝑘 ∈ (𝑦𝐻𝑋)𝑛 = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩𝑂𝑍)((𝑦𝐺𝑋)‘𝑘)) ↔ ∃!𝑘 ∈ (𝑌𝐻𝑋)𝑛 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑘))))
14	4, 13	raleqbidv 3323	. . . 4 ⊢ (𝑦 = 𝑌 → (∀𝑛 ∈ ((𝐹‘𝑦)𝐽𝑍)∃!𝑘 ∈ (𝑦𝐻𝑋)𝑛 = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩𝑂𝑍)((𝑦𝐺𝑋)‘𝑘)) ↔ ∀𝑛 ∈ ((𝐹‘𝑌)𝐽𝑍)∃!𝑘 ∈ (𝑌𝐻𝑋)𝑛 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑘))))
15		oppcup3lem.1	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑛 ∈ ((𝐹‘𝑦)𝐽𝑍)∃!𝑘 ∈ (𝑦𝐻𝑋)𝑛 = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩𝑂𝑍)((𝑦𝐺𝑋)‘𝑘)))
16		oppcup3lem.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
17	14, 15, 16	rspcdva 3600	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ ((𝐹‘𝑌)𝐽𝑍)∃!𝑘 ∈ (𝑌𝐻𝑋)𝑛 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑘)))
18		oppcup3lem.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ((𝐹‘𝑌)𝐽𝑍))
19	2, 17, 18	rspcdva 3600	. 2 ⊢ (𝜑 → ∃!𝑘 ∈ (𝑌𝐻𝑋)𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑘)))
20		fveq2 6873	. . . . . 6 ⊢ (𝑘 = 𝑚 → ((𝑌𝐺𝑋)‘𝑘) = ((𝑌𝐺𝑋)‘𝑚))
21	20	oveq2d 7416	. . . . 5 ⊢ (𝑘 = 𝑚 → (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑘)) = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑚)))
22	21	eqeq2d 2745	. . . 4 ⊢ (𝑘 = 𝑚 → (𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑘)) ↔ 𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑚))))
23	22	cbvreuvw 3381	. . 3 ⊢ (∃!𝑘 ∈ (𝑌𝐻𝑋)𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑘)) ↔ ∃!𝑚 ∈ (𝑌𝐻𝑋)𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑚)))
24		fveq2 6873	. . . . . 6 ⊢ (𝑚 = 𝑙 → ((𝑌𝐺𝑋)‘𝑚) = ((𝑌𝐺𝑋)‘𝑙))
25	24	oveq2d 7416	. . . . 5 ⊢ (𝑚 = 𝑙 → (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑚)) = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑙)))
26	25	eqeq2d 2745	. . . 4 ⊢ (𝑚 = 𝑙 → (𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑚)) ↔ 𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑙))))
27	26	cbvreuvw 3381	. . 3 ⊢ (∃!𝑚 ∈ (𝑌𝐻𝑋)𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑚)) ↔ ∃!𝑙 ∈ (𝑌𝐻𝑋)𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑙)))
28	23, 27	bitri 275	. 2 ⊢ (∃!𝑘 ∈ (𝑌𝐻𝑋)𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑘)) ↔ ∃!𝑙 ∈ (𝑌𝐻𝑋)𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑙)))
29	19, 28	sylib 218	1 ⊢ (𝜑 → ∃!𝑙 ∈ (𝑌𝐻𝑋)𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑙)))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  ∀wral 3050  ∃!wreu 3355  ⟨cop 4605  ‘cfv 6528  (class class class)co 7400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-reu 3358  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-br 5118  df-iota 6481  df-fv 6536  df-ov 7403

This theorem is referenced by:  oppcup3  49005