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Theorem dmdbr 29142
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 2688 . . . . 5 (𝑦 = 𝐴 → (𝑦C𝐴C ))
21anbi1d 741 . . . 4 (𝑦 = 𝐴 → ((𝑦C𝑧C ) ↔ (𝐴C𝑧C )))
3 ineq2 3806 . . . . . . . 8 (𝑦 = 𝐴 → (𝑥𝑦) = (𝑥𝐴))
43oveq1d 6662 . . . . . . 7 (𝑦 = 𝐴 → ((𝑥𝑦) ∨ 𝑧) = ((𝑥𝐴) ∨ 𝑧))
5 oveq1 6654 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦 𝑧) = (𝐴 𝑧))
65ineq2d 3812 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 ∩ (𝑦 𝑧)) = (𝑥 ∩ (𝐴 𝑧)))
74, 6eqeq12d 2636 . . . . . 6 (𝑦 = 𝐴 → (((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧)) ↔ ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧))))
87imbi2d 330 . . . . 5 (𝑦 = 𝐴 → ((𝑧𝑥 → ((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧))) ↔ (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧)))))
98ralbidv 2985 . . . 4 (𝑦 = 𝐴 → (∀𝑥C (𝑧𝑥 → ((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧))) ↔ ∀𝑥C (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧)))))
102, 9anbi12d 747 . . 3 (𝑦 = 𝐴 → (((𝑦C𝑧C ) ∧ ∀𝑥C (𝑧𝑥 → ((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧)))) ↔ ((𝐴C𝑧C ) ∧ ∀𝑥C (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧))))))
11 eleq1 2688 . . . . 5 (𝑧 = 𝐵 → (𝑧C𝐵C ))
1211anbi2d 740 . . . 4 (𝑧 = 𝐵 → ((𝐴C𝑧C ) ↔ (𝐴C𝐵C )))
13 sseq1 3624 . . . . . 6 (𝑧 = 𝐵 → (𝑧𝑥𝐵𝑥))
14 oveq2 6655 . . . . . . 7 (𝑧 = 𝐵 → ((𝑥𝐴) ∨ 𝑧) = ((𝑥𝐴) ∨ 𝐵))
15 oveq2 6655 . . . . . . . 8 (𝑧 = 𝐵 → (𝐴 𝑧) = (𝐴 𝐵))
1615ineq2d 3812 . . . . . . 7 (𝑧 = 𝐵 → (𝑥 ∩ (𝐴 𝑧)) = (𝑥 ∩ (𝐴 𝐵)))
1714, 16eqeq12d 2636 . . . . . 6 (𝑧 = 𝐵 → (((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧)) ↔ ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵))))
1813, 17imbi12d 334 . . . . 5 (𝑧 = 𝐵 → ((𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧))) ↔ (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
1918ralbidv 2985 . . . 4 (𝑧 = 𝐵 → (∀𝑥C (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧))) ↔ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
2012, 19anbi12d 747 . . 3 (𝑧 = 𝐵 → (((𝐴C𝑧C ) ∧ ∀𝑥C (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧)))) ↔ ((𝐴C𝐵C ) ∧ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵))))))
21 df-dmd 29124 . . 3 𝑀* = {⟨𝑦, 𝑧⟩ ∣ ((𝑦C𝑧C ) ∧ ∀𝑥C (𝑧𝑥 → ((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧))))}
2210, 20, 21brabg 4992 . 2 ((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 ↔ ((𝐴C𝐵C ) ∧ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵))))))
2322bianabs 924 1 ((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 ↔ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  wral 2911  cin 3571  wss 3572   class class class wbr 4651  (class class class)co 6647   C cch 27770   chj 27774   𝑀* cdmd 27808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-iota 5849  df-fv 5894  df-ov 6650  df-dmd 29124
This theorem is referenced by:  dmdmd  29143  dmdi  29145  dmdbr2  29146  dmdbr3  29148  mddmd2  29152
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