Theorem dmdbr 32331

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 2832	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ Cℋ ↔ 𝐴 ∈ Cℋ ))
2	1	anbi1d 630	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ )))
3		ineq2 4235	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑥 ∩ 𝑦) = (𝑥 ∩ 𝐴))
4	3	oveq1d 7463	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝑥 ∩ 𝑦) ∨ℋ 𝑧) = ((𝑥 ∩ 𝐴) ∨ℋ 𝑧))
5		oveq1 7455	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 ∨ℋ 𝑧) = (𝐴 ∨ℋ 𝑧))
6	5	ineq2d 4241	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ∩ (𝑦 ∨ℋ 𝑧)) = (𝑥 ∩ (𝐴 ∨ℋ 𝑧)))
7	4, 6	eqeq12d 2756	. . . . . 6 ⊢ (𝑦 = 𝐴 → (((𝑥 ∩ 𝑦) ∨ℋ 𝑧) = (𝑥 ∩ (𝑦 ∨ℋ 𝑧)) ↔ ((𝑥 ∩ 𝐴) ∨ℋ 𝑧) = (𝑥 ∩ (𝐴 ∨ℋ 𝑧))))
8	7	imbi2d 340	. . . . 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 3184	. . . 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 631	. . 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 2832	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ Cℋ ↔ 𝐵 ∈ Cℋ ))
12	11	anbi2d 629	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ )))
13		sseq1 4034	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑧 ⊆ 𝑥 ↔ 𝐵 ⊆ 𝑥))
14		oveq2 7456	. . . . . . 7 ⊢ (𝑧 = 𝐵 → ((𝑥 ∩ 𝐴) ∨ℋ 𝑧) = ((𝑥 ∩ 𝐴) ∨ℋ 𝐵))
15		oveq2 7456	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝐴 ∨ℋ 𝑧) = (𝐴 ∨ℋ 𝐵))
16	15	ineq2d 4241	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝑥 ∩ (𝐴 ∨ℋ 𝑧)) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵)))
17	14, 16	eqeq12d 2756	. . . . . 6 ⊢ (𝑧 = 𝐵 → (((𝑥 ∩ 𝐴) ∨ℋ 𝑧) = (𝑥 ∩ (𝐴 ∨ℋ 𝑧)) ↔ ((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵))))
18	13, 17	imbi12d 344	. . . . 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 3184	. . . 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 631	. . 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 32313	. . 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 5558	. 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 541	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 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ∩ cin 3975   ⊆ wss 3976   class class class wbr 5166  (class class class)co 7448   C_ℋ cch 30961   ∨_ℋ chj 30965   𝑀_ℋ^* cdmd 30999

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  ax-sep 5317  ax-nul 5324  ax-pr 5447

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-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-iota 6525  df-fv 6581  df-ov 7451  df-dmd 32313

This theorem is referenced by:  dmdmd  32332  dmdi  32334  dmdbr2  32335  dmdbr3  32337  mddmd2  32341