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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 ((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 ↔ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 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.
StepHypRef Expression
1 eleq1 2676 . . . . 5 (𝑦 = 𝐴 → (𝑦C𝐴C ))
21anbi1d 737 . . . 4 (𝑦 = 𝐴 → ((𝑦C𝑧C ) ↔ (𝐴C𝑧C )))
3 ineq2 3770 . . . . . . . 8 (𝑦 = 𝐴 → (𝑥𝑦) = (𝑥𝐴))
43oveq1d 6564 . . . . . . 7 (𝑦 = 𝐴 → ((𝑥𝑦) ∨ 𝑧) = ((𝑥𝐴) ∨ 𝑧))
5 oveq1 6556 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦 𝑧) = (𝐴 𝑧))
65ineq2d 3776 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 ∩ (𝑦 𝑧)) = (𝑥 ∩ (𝐴 𝑧)))
74, 6eqeq12d 2625 . . . . . 6 (𝑦 = 𝐴 → (((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧)) ↔ ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧))))
87imbi2d 329 . . . . 5 (𝑦 = 𝐴 → ((𝑧𝑥 → ((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧))) ↔ (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧)))))
98ralbidv 2969 . . . 4 (𝑦 = 𝐴 → (∀𝑥C (𝑧𝑥 → ((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧))) ↔ ∀𝑥C (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧)))))
102, 9anbi12d 743 . . 3 (𝑦 = 𝐴 → (((𝑦C𝑧C ) ∧ ∀𝑥C (𝑧𝑥 → ((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧)))) ↔ ((𝐴C𝑧C ) ∧ ∀𝑥C (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧))))))
11 eleq1 2676 . . . . 5 (𝑧 = 𝐵 → (𝑧C𝐵C ))
1211anbi2d 736 . . . 4 (𝑧 = 𝐵 → ((𝐴C𝑧C ) ↔ (𝐴C𝐵C )))
13 sseq1 3589 . . . . . 6 (𝑧 = 𝐵 → (𝑧𝑥𝐵𝑥))
14 oveq2 6557 . . . . . . 7 (𝑧 = 𝐵 → ((𝑥𝐴) ∨ 𝑧) = ((𝑥𝐴) ∨ 𝐵))
15 oveq2 6557 . . . . . . . 8 (𝑧 = 𝐵 → (𝐴 𝑧) = (𝐴 𝐵))
1615ineq2d 3776 . . . . . . 7 (𝑧 = 𝐵 → (𝑥 ∩ (𝐴 𝑧)) = (𝑥 ∩ (𝐴 𝐵)))
1714, 16eqeq12d 2625 . . . . . 6 (𝑧 = 𝐵 → (((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧)) ↔ ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵))))
1813, 17imbi12d 333 . . . . 5 (𝑧 = 𝐵 → ((𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧))) ↔ (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
1918ralbidv 2969 . . . 4 (𝑧 = 𝐵 → (∀𝑥C (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧))) ↔ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
2012, 19anbi12d 743 . . 3 (𝑧 = 𝐵 → (((𝐴C𝑧C ) ∧ ∀𝑥C (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧)))) ↔ ((𝐴C𝐵C ) ∧ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵))))))
21 df-dmd 28524 . . 3 𝑀* = {⟨𝑦, 𝑧⟩ ∣ ((𝑦C𝑧C ) ∧ ∀𝑥C (𝑧𝑥 → ((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧))))}
2210, 20, 21brabg 4919 . 2 ((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 ↔ ((𝐴C𝐵C ) ∧ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵))))))
2322bianabs 920 1 ((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 ↔ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
 Colors of variables: wff setvar class 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   Cℋ cch 27170   ∨ℋ chj 27174   𝑀ℋ* 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
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