Theorem dmdbr 30079

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 2903	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ Cℋ ↔ 𝐴 ∈ Cℋ ))
2	1	anbi1d 631	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ )))
3		ineq2 4186	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑥 ∩ 𝑦) = (𝑥 ∩ 𝐴))
4	3	oveq1d 7174	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝑥 ∩ 𝑦) ∨ℋ 𝑧) = ((𝑥 ∩ 𝐴) ∨ℋ 𝑧))
5		oveq1 7166	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 ∨ℋ 𝑧) = (𝐴 ∨ℋ 𝑧))
6	5	ineq2d 4192	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ∩ (𝑦 ∨ℋ 𝑧)) = (𝑥 ∩ (𝐴 ∨ℋ 𝑧)))
7	4, 6	eqeq12d 2840	. . . . . 6 ⊢ (𝑦 = 𝐴 → (((𝑥 ∩ 𝑦) ∨ℋ 𝑧) = (𝑥 ∩ (𝑦 ∨ℋ 𝑧)) ↔ ((𝑥 ∩ 𝐴) ∨ℋ 𝑧) = (𝑥 ∩ (𝐴 ∨ℋ 𝑧))))
8	7	imbi2d 343	. . . . 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 3200	. . . 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 632	. . 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 2903	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ Cℋ ↔ 𝐵 ∈ Cℋ ))
12	11	anbi2d 630	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ )))
13		sseq1 3995	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑧 ⊆ 𝑥 ↔ 𝐵 ⊆ 𝑥))
14		oveq2 7167	. . . . . . 7 ⊢ (𝑧 = 𝐵 → ((𝑥 ∩ 𝐴) ∨ℋ 𝑧) = ((𝑥 ∩ 𝐴) ∨ℋ 𝐵))
15		oveq2 7167	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝐴 ∨ℋ 𝑧) = (𝐴 ∨ℋ 𝐵))
16	15	ineq2d 4192	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝑥 ∩ (𝐴 ∨ℋ 𝑧)) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵)))
17	14, 16	eqeq12d 2840	. . . . . 6 ⊢ (𝑧 = 𝐵 → (((𝑥 ∩ 𝐴) ∨ℋ 𝑧) = (𝑥 ∩ (𝐴 ∨ℋ 𝑧)) ↔ ((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵))))
18	13, 17	imbi12d 347	. . . . 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 3200	. . . 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 632	. . 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 30061	. . 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 5429	. 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 544	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 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ∀wral 3141   ∩ cin 3938   ⊆ wss 3939   class class class wbr 5069  (class class class)co 7159   C_ℋ cch 28709   ∨_ℋ chj 28713   𝑀_ℋ^* cdmd 28747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-iota 6317  df-fv 6366  df-ov 7162  df-dmd 30061

This theorem is referenced by:  dmdmd  30080  dmdi  30082  dmdbr2  30083  dmdbr3  30085  mddmd2  30089