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Theorem dmdbr 32328
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 2827 . . . . 5 (𝑦 = 𝐴 → (𝑦C𝐴C ))
21anbi1d 631 . . . 4 (𝑦 = 𝐴 → ((𝑦C𝑧C ) ↔ (𝐴C𝑧C )))
3 ineq2 4222 . . . . . . . 8 (𝑦 = 𝐴 → (𝑥𝑦) = (𝑥𝐴))
43oveq1d 7446 . . . . . . 7 (𝑦 = 𝐴 → ((𝑥𝑦) ∨ 𝑧) = ((𝑥𝐴) ∨ 𝑧))
5 oveq1 7438 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦 𝑧) = (𝐴 𝑧))
65ineq2d 4228 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 ∩ (𝑦 𝑧)) = (𝑥 ∩ (𝐴 𝑧)))
74, 6eqeq12d 2751 . . . . . 6 (𝑦 = 𝐴 → (((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧)) ↔ ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧))))
87imbi2d 340 . . . . 5 (𝑦 = 𝐴 → ((𝑧𝑥 → ((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧))) ↔ (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧)))))
98ralbidv 3176 . . . 4 (𝑦 = 𝐴 → (∀𝑥C (𝑧𝑥 → ((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧))) ↔ ∀𝑥C (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧)))))
102, 9anbi12d 632 . . 3 (𝑦 = 𝐴 → (((𝑦C𝑧C ) ∧ ∀𝑥C (𝑧𝑥 → ((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧)))) ↔ ((𝐴C𝑧C ) ∧ ∀𝑥C (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧))))))
11 eleq1 2827 . . . . 5 (𝑧 = 𝐵 → (𝑧C𝐵C ))
1211anbi2d 630 . . . 4 (𝑧 = 𝐵 → ((𝐴C𝑧C ) ↔ (𝐴C𝐵C )))
13 sseq1 4021 . . . . . 6 (𝑧 = 𝐵 → (𝑧𝑥𝐵𝑥))
14 oveq2 7439 . . . . . . 7 (𝑧 = 𝐵 → ((𝑥𝐴) ∨ 𝑧) = ((𝑥𝐴) ∨ 𝐵))
15 oveq2 7439 . . . . . . . 8 (𝑧 = 𝐵 → (𝐴 𝑧) = (𝐴 𝐵))
1615ineq2d 4228 . . . . . . 7 (𝑧 = 𝐵 → (𝑥 ∩ (𝐴 𝑧)) = (𝑥 ∩ (𝐴 𝐵)))
1714, 16eqeq12d 2751 . . . . . 6 (𝑧 = 𝐵 → (((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧)) ↔ ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵))))
1813, 17imbi12d 344 . . . . 5 (𝑧 = 𝐵 → ((𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧))) ↔ (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
1918ralbidv 3176 . . . 4 (𝑧 = 𝐵 → (∀𝑥C (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧))) ↔ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
2012, 19anbi12d 632 . . 3 (𝑧 = 𝐵 → (((𝐴C𝑧C ) ∧ ∀𝑥C (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧)))) ↔ ((𝐴C𝐵C ) ∧ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵))))))
21 df-dmd 32310 . . 3 𝑀* = {⟨𝑦, 𝑧⟩ ∣ ((𝑦C𝑧C ) ∧ ∀𝑥C (𝑧𝑥 → ((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧))))}
2210, 20, 21brabg 5549 . 2 ((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 ↔ ((𝐴C𝐵C ) ∧ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵))))))
2322bianabs 541 1 ((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 ↔ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  cin 3962  wss 3963   class class class wbr 5148  (class class class)co 7431   C cch 30958   chj 30962   𝑀* cdmd 30996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-iota 6516  df-fv 6571  df-ov 7434  df-dmd 32310
This theorem is referenced by:  dmdmd  32329  dmdi  32331  dmdbr2  32332  dmdbr3  32334  mddmd2  32338
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