Theorem dmdbr 32901

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 2848	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ Cℋ ↔ 𝐴 ∈ Cℋ ))
2	1	anbi1d 643	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ )))
3		ineq2 4111	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑥 ∩ 𝑦) = (𝑥 ∩ 𝐴))
4	3	oveq1d 7435	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝑥 ∩ 𝑦) ∨ℋ 𝑧) = ((𝑥 ∩ 𝐴) ∨ℋ 𝑧))
5		oveq1 7427	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 ∨ℋ 𝑧) = (𝐴 ∨ℋ 𝑧))
6	5	ineq2d 4117	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ∩ (𝑦 ∨ℋ 𝑧)) = (𝑥 ∩ (𝐴 ∨ℋ 𝑧)))
7	4, 6	eqeq12d 2776	. . . . . 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 3185	. . . 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 644	. . 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 2848	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ Cℋ ↔ 𝐵 ∈ Cℋ ))
12	11	anbi2d 642	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝐴 ∈ Cℋ ∧ 𝑧 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ )))
13		sseq1 3933	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑧 ⊆ 𝑥 ↔ 𝐵 ⊆ 𝑥))
14		oveq2 7428	. . . . . . 7 ⊢ (𝑧 = 𝐵 → ((𝑥 ∩ 𝐴) ∨ℋ 𝑧) = ((𝑥 ∩ 𝐴) ∨ℋ 𝐵))
15		oveq2 7428	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝐴 ∨ℋ 𝑧) = (𝐴 ∨ℋ 𝐵))
16	15	ineq2d 4117	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝑥 ∩ (𝐴 ∨ℋ 𝑧)) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵)))
17	14, 16	eqeq12d 2776	. . . . . 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 3185	. . . 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 644	. . 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 32883	. . 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 5513	. 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 551	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑥 ∈ Cℋ (𝐵 ⊆ 𝑥 → ((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895   ∩ cin 4089   class class class wbr 5102  (class class class)co 7420   C_ℋ cch 31531   ∨_ℋ chj 31535   𝑀_ℋ^* cdmd 31569

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-iota 6494  df-fv 6546  df-ov 7423  df-dmd 32883

This theorem is used by:  dmdmd  32902  dmdi  32904  dmdbr2  32905  dmdbr3  32907  mddmd2  32911