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Theorem mdbr 32097
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 2817 . . . . 5 (𝑦 = 𝐴 → (𝑦C𝐴C ))
21anbi1d 630 . . . 4 (𝑦 = 𝐴 → ((𝑦C𝑧C ) ↔ (𝐴C𝑧C )))
3 oveq2 7422 . . . . . . . 8 (𝑦 = 𝐴 → (𝑥 𝑦) = (𝑥 𝐴))
43ineq1d 4207 . . . . . . 7 (𝑦 = 𝐴 → ((𝑥 𝑦) ∩ 𝑧) = ((𝑥 𝐴) ∩ 𝑧))
5 ineq1 4201 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦𝑧) = (𝐴𝑧))
65oveq2d 7430 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 (𝑦𝑧)) = (𝑥 (𝐴𝑧)))
74, 6eqeq12d 2744 . . . . . 6 (𝑦 = 𝐴 → (((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧)) ↔ ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧))))
87imbi2d 340 . . . . 5 (𝑦 = 𝐴 → ((𝑥𝑧 → ((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧))) ↔ (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧)))))
98ralbidv 3173 . . . 4 (𝑦 = 𝐴 → (∀𝑥C (𝑥𝑧 → ((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧))) ↔ ∀𝑥C (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧)))))
102, 9anbi12d 631 . . 3 (𝑦 = 𝐴 → (((𝑦C𝑧C ) ∧ ∀𝑥C (𝑥𝑧 → ((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧)))) ↔ ((𝐴C𝑧C ) ∧ ∀𝑥C (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧))))))
11 eleq1 2817 . . . . 5 (𝑧 = 𝐵 → (𝑧C𝐵C ))
1211anbi2d 629 . . . 4 (𝑧 = 𝐵 → ((𝐴C𝑧C ) ↔ (𝐴C𝐵C )))
13 sseq2 4004 . . . . . 6 (𝑧 = 𝐵 → (𝑥𝑧𝑥𝐵))
14 ineq2 4202 . . . . . . 7 (𝑧 = 𝐵 → ((𝑥 𝐴) ∩ 𝑧) = ((𝑥 𝐴) ∩ 𝐵))
15 ineq2 4202 . . . . . . . 8 (𝑧 = 𝐵 → (𝐴𝑧) = (𝐴𝐵))
1615oveq2d 7430 . . . . . . 7 (𝑧 = 𝐵 → (𝑥 (𝐴𝑧)) = (𝑥 (𝐴𝐵)))
1714, 16eqeq12d 2744 . . . . . 6 (𝑧 = 𝐵 → (((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧)) ↔ ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))))
1813, 17imbi12d 344 . . . . 5 (𝑧 = 𝐵 → ((𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧))) ↔ (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))))
1918ralbidv 3173 . . . 4 (𝑧 = 𝐵 → (∀𝑥C (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧))) ↔ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))))
2012, 19anbi12d 631 . . 3 (𝑧 = 𝐵 → (((𝐴C𝑧C ) ∧ ∀𝑥C (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧)))) ↔ ((𝐴C𝐵C ) ∧ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))))))
21 df-md 32083 . . 3 𝑀 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦C𝑧C ) ∧ ∀𝑥C (𝑥𝑧 → ((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧))))}
2210, 20, 21brabg 5535 . 2 ((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ((𝐴C𝐵C ) ∧ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))))))
2322bianabs 541 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 395   = wceq 1534  wcel 2099  wral 3057  cin 3944  wss 3945   class class class wbr 5142  (class class class)co 7414   C cch 30732   chj 30736   𝑀 cmd 30769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-iota 6494  df-fv 6550  df-ov 7417  df-md 32083
This theorem is referenced by:  mdi  32098  mdbr2  32099  mdbr3  32100  dmdmd  32103  mddmd2  32112  mdsl1i  32124
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