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Theorem mdbr 31524
Description: Binary relation expressing 𝐴, 𝐵 is a modular pair. Definition 1.1 of [MaedaMaeda] p. 1. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
mdbr ((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem mdbr
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2822 . . . . 5 (𝑦 = 𝐴 → (𝑦C𝐴C ))
21anbi1d 631 . . . 4 (𝑦 = 𝐴 → ((𝑦C𝑧C ) ↔ (𝐴C𝑧C )))
3 oveq2 7411 . . . . . . . 8 (𝑦 = 𝐴 → (𝑥 𝑦) = (𝑥 𝐴))
43ineq1d 4209 . . . . . . 7 (𝑦 = 𝐴 → ((𝑥 𝑦) ∩ 𝑧) = ((𝑥 𝐴) ∩ 𝑧))
5 ineq1 4203 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦𝑧) = (𝐴𝑧))
65oveq2d 7419 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 (𝑦𝑧)) = (𝑥 (𝐴𝑧)))
74, 6eqeq12d 2749 . . . . . 6 (𝑦 = 𝐴 → (((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧)) ↔ ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧))))
87imbi2d 341 . . . . 5 (𝑦 = 𝐴 → ((𝑥𝑧 → ((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧))) ↔ (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧)))))
98ralbidv 3178 . . . 4 (𝑦 = 𝐴 → (∀𝑥C (𝑥𝑧 → ((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧))) ↔ ∀𝑥C (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧)))))
102, 9anbi12d 632 . . 3 (𝑦 = 𝐴 → (((𝑦C𝑧C ) ∧ ∀𝑥C (𝑥𝑧 → ((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧)))) ↔ ((𝐴C𝑧C ) ∧ ∀𝑥C (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧))))))
11 eleq1 2822 . . . . 5 (𝑧 = 𝐵 → (𝑧C𝐵C ))
1211anbi2d 630 . . . 4 (𝑧 = 𝐵 → ((𝐴C𝑧C ) ↔ (𝐴C𝐵C )))
13 sseq2 4006 . . . . . 6 (𝑧 = 𝐵 → (𝑥𝑧𝑥𝐵))
14 ineq2 4204 . . . . . . 7 (𝑧 = 𝐵 → ((𝑥 𝐴) ∩ 𝑧) = ((𝑥 𝐴) ∩ 𝐵))
15 ineq2 4204 . . . . . . . 8 (𝑧 = 𝐵 → (𝐴𝑧) = (𝐴𝐵))
1615oveq2d 7419 . . . . . . 7 (𝑧 = 𝐵 → (𝑥 (𝐴𝑧)) = (𝑥 (𝐴𝐵)))
1714, 16eqeq12d 2749 . . . . . 6 (𝑧 = 𝐵 → (((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧)) ↔ ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))))
1813, 17imbi12d 345 . . . . 5 (𝑧 = 𝐵 → ((𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧))) ↔ (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))))
1918ralbidv 3178 . . . 4 (𝑧 = 𝐵 → (∀𝑥C (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧))) ↔ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))))
2012, 19anbi12d 632 . . 3 (𝑧 = 𝐵 → (((𝐴C𝑧C ) ∧ ∀𝑥C (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧)))) ↔ ((𝐴C𝐵C ) ∧ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))))))
21 df-md 31510 . . 3 𝑀 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦C𝑧C ) ∧ ∀𝑥C (𝑥𝑧 → ((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧))))}
2210, 20, 21brabg 5537 . 2 ((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ((𝐴C𝐵C ) ∧ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))))))
2322bianabs 543 1 ((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  cin 3945  wss 3946   class class class wbr 5146  (class class class)co 7403   C cch 30159   chj 30163   𝑀 cmd 30196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5297  ax-nul 5304  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-br 5147  df-opab 5209  df-iota 6491  df-fv 6547  df-ov 7406  df-md 31510
This theorem is referenced by:  mdi  31525  mdbr2  31526  mdbr3  31527  dmdmd  31530  mddmd2  31539  mdsl1i  31551
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