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Theorem mdbr 28371
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 2675 . . . . 5 (𝑦 = 𝐴 → (𝑦C𝐴C ))
21anbi1d 736 . . . 4 (𝑦 = 𝐴 → ((𝑦C𝑧C ) ↔ (𝐴C𝑧C )))
3 oveq2 6535 . . . . . . . 8 (𝑦 = 𝐴 → (𝑥 𝑦) = (𝑥 𝐴))
43ineq1d 3774 . . . . . . 7 (𝑦 = 𝐴 → ((𝑥 𝑦) ∩ 𝑧) = ((𝑥 𝐴) ∩ 𝑧))
5 ineq1 3768 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦𝑧) = (𝐴𝑧))
65oveq2d 6543 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 (𝑦𝑧)) = (𝑥 (𝐴𝑧)))
74, 6eqeq12d 2624 . . . . . 6 (𝑦 = 𝐴 → (((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧)) ↔ ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧))))
87imbi2d 328 . . . . 5 (𝑦 = 𝐴 → ((𝑥𝑧 → ((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧))) ↔ (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧)))))
98ralbidv 2968 . . . 4 (𝑦 = 𝐴 → (∀𝑥C (𝑥𝑧 → ((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧))) ↔ ∀𝑥C (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧)))))
102, 9anbi12d 742 . . 3 (𝑦 = 𝐴 → (((𝑦C𝑧C ) ∧ ∀𝑥C (𝑥𝑧 → ((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧)))) ↔ ((𝐴C𝑧C ) ∧ ∀𝑥C (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧))))))
11 eleq1 2675 . . . . 5 (𝑧 = 𝐵 → (𝑧C𝐵C ))
1211anbi2d 735 . . . 4 (𝑧 = 𝐵 → ((𝐴C𝑧C ) ↔ (𝐴C𝐵C )))
13 sseq2 3589 . . . . . 6 (𝑧 = 𝐵 → (𝑥𝑧𝑥𝐵))
14 ineq2 3769 . . . . . . 7 (𝑧 = 𝐵 → ((𝑥 𝐴) ∩ 𝑧) = ((𝑥 𝐴) ∩ 𝐵))
15 ineq2 3769 . . . . . . . 8 (𝑧 = 𝐵 → (𝐴𝑧) = (𝐴𝐵))
1615oveq2d 6543 . . . . . . 7 (𝑧 = 𝐵 → (𝑥 (𝐴𝑧)) = (𝑥 (𝐴𝐵)))
1714, 16eqeq12d 2624 . . . . . 6 (𝑧 = 𝐵 → (((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧)) ↔ ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))))
1813, 17imbi12d 332 . . . . 5 (𝑧 = 𝐵 → ((𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧))) ↔ (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))))
1918ralbidv 2968 . . . 4 (𝑧 = 𝐵 → (∀𝑥C (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧))) ↔ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))))
2012, 19anbi12d 742 . . 3 (𝑧 = 𝐵 → (((𝐴C𝑧C ) ∧ ∀𝑥C (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧)))) ↔ ((𝐴C𝐵C ) ∧ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))))))
21 df-md 28357 . . 3 𝑀 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦C𝑧C ) ∧ ∀𝑥C (𝑥𝑧 → ((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧))))}
2210, 20, 21brabg 4909 . 2 ((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ((𝐴C𝐵C ) ∧ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))))))
2322bianabs 919 1 ((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2895  cin 3538  wss 3539   class class class wbr 4577  (class class class)co 6527   C cch 27004   chj 27008   𝑀 cmd 27041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-iota 5754  df-fv 5798  df-ov 6530  df-md 28357
This theorem is referenced by:  mdi  28372  mdbr2  28373  mdbr3  28374  dmdmd  28377  mddmd2  28386  mdsl1i  28398
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