Theorem dmdbr 28542

Description: Binary relation expressing the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
dmdbr	⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑥 ∈ Cℋ (𝐵 ⊆ 𝑥 → ((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵)))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dmdbr

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2676	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ Cℋ ↔ 𝐴 ∈ Cℋ ))
2	1	anbi1d 737	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ )))
3		ineq2 3770	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑥 ∩ 𝑦) = (𝑥 ∩ 𝐴))
4	3	oveq1d 6564	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝑥 ∩ 𝑦) ∨ℋ 𝑧) = ((𝑥 ∩ 𝐴) ∨ℋ 𝑧))
5		oveq1 6556	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 ∨ℋ 𝑧) = (𝐴 ∨ℋ 𝑧))
6	5	ineq2d 3776	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ∩ (𝑦 ∨ℋ 𝑧)) = (𝑥 ∩ (𝐴 ∨ℋ 𝑧)))
7	4, 6	eqeq12d 2625	. . . . . 6 ⊢ (𝑦 = 𝐴 → (((𝑥 ∩ 𝑦) ∨ℋ 𝑧) = (𝑥 ∩ (𝑦 ∨ℋ 𝑧)) ↔ ((𝑥 ∩ 𝐴) ∨ℋ 𝑧) = (𝑥 ∩ (𝐴 ∨ℋ 𝑧))))
8	7	imbi2d 329	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑧 ⊆ 𝑥 → ((𝑥 ∩ 𝑦) ∨ℋ 𝑧) = (𝑥 ∩ (𝑦 ∨ℋ 𝑧))) ↔ (𝑧 ⊆ 𝑥 → ((𝑥 ∩ 𝐴) ∨ℋ 𝑧) = (𝑥 ∩ (𝐴 ∨ℋ 𝑧)))))
9	8	ralbidv 2969	. . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ Cℋ (𝑧 ⊆ 𝑥 → ((𝑥 ∩ 𝑦) ∨ℋ 𝑧) = (𝑥 ∩ (𝑦 ∨ℋ 𝑧))) ↔ ∀𝑥 ∈ Cℋ (𝑧 ⊆ 𝑥 → ((𝑥 ∩ 𝐴) ∨ℋ 𝑧) = (𝑥 ∩ (𝐴 ∨ℋ 𝑧)))))
10	2, 9	anbi12d 743	. . 3 ⊢ (𝑦 = 𝐴 → (((𝑦 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ∧ ∀𝑥 ∈ Cℋ (𝑧 ⊆ 𝑥 → ((𝑥 ∩ 𝑦) ∨ℋ 𝑧) = (𝑥 ∩ (𝑦 ∨ℋ 𝑧)))) ↔ ((𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ∧ ∀𝑥 ∈ Cℋ (𝑧 ⊆ 𝑥 → ((𝑥 ∩ 𝐴) ∨ℋ 𝑧) = (𝑥 ∩ (𝐴 ∨ℋ 𝑧))))))
11		eleq1 2676	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ Cℋ ↔ 𝐵 ∈ Cℋ ))
12	11	anbi2d 736	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ )))
13		sseq1 3589	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑧 ⊆ 𝑥 ↔ 𝐵 ⊆ 𝑥))
14		oveq2 6557	. . . . . . 7 ⊢ (𝑧 = 𝐵 → ((𝑥 ∩ 𝐴) ∨ℋ 𝑧) = ((𝑥 ∩ 𝐴) ∨ℋ 𝐵))
15		oveq2 6557	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝐴 ∨ℋ 𝑧) = (𝐴 ∨ℋ 𝐵))
16	15	ineq2d 3776	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝑥 ∩ (𝐴 ∨ℋ 𝑧)) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵)))
17	14, 16	eqeq12d 2625	. . . . . 6 ⊢ (𝑧 = 𝐵 → (((𝑥 ∩ 𝐴) ∨ℋ 𝑧) = (𝑥 ∩ (𝐴 ∨ℋ 𝑧)) ↔ ((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵))))
18	13, 17	imbi12d 333	. . . . 5 ⊢ (𝑧 = 𝐵 → ((𝑧 ⊆ 𝑥 → ((𝑥 ∩ 𝐴) ∨ℋ 𝑧) = (𝑥 ∩ (𝐴 ∨ℋ 𝑧))) ↔ (𝐵 ⊆ 𝑥 → ((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵)))))
19	18	ralbidv 2969	. . . 4 ⊢ (𝑧 = 𝐵 → (∀𝑥 ∈ Cℋ (𝑧 ⊆ 𝑥 → ((𝑥 ∩ 𝐴) ∨ℋ 𝑧) = (𝑥 ∩ (𝐴 ∨ℋ 𝑧))) ↔ ∀𝑥 ∈ Cℋ (𝐵 ⊆ 𝑥 → ((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵)))))
20	12, 19	anbi12d 743	. . 3 ⊢ (𝑧 = 𝐵 → (((𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ∧ ∀𝑥 ∈ Cℋ (𝑧 ⊆ 𝑥 → ((𝑥 ∩ 𝐴) ∨ℋ 𝑧) = (𝑥 ∩ (𝐴 ∨ℋ 𝑧)))) ↔ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ ∀𝑥 ∈ Cℋ (𝐵 ⊆ 𝑥 → ((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵))))))
21		df-dmd 28524	. . 3 ⊢ 𝑀ℋ* = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ∧ ∀𝑥 ∈ Cℋ (𝑧 ⊆ 𝑥 → ((𝑥 ∩ 𝑦) ∨ℋ 𝑧) = (𝑥 ∩ (𝑦 ∨ℋ 𝑧))))}
22	10, 20, 21	brabg 4919	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 ↔ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ ∀𝑥 ∈ Cℋ (𝐵 ⊆ 𝑥 → ((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵))))))
23	22	bianabs 920	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑥 ∈ Cℋ (𝐵 ⊆ 𝑥 → ((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵)))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∩ cin 3539   ⊆ wss 3540   class class class wbr 4583  (class class class)co 6549   C_ℋ cch 27170   ∨_ℋ chj 27174   𝑀_ℋ^* cdmd 27208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-iota 5768  df-fv 5812  df-ov 6552  df-dmd 28524

This theorem is referenced by:  dmdmd  28543  dmdi  28545  dmdbr2  28546  dmdbr3  28548  mddmd2  28552