ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  soinxp GIF version

Theorem soinxp 4466
Description: Intersection of linear order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
Assertion
Ref Expression
soinxp (𝑅 Or 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴)

Proof of Theorem soinxp
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poinxp 4465 . . 3 (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)
2 brinxp 4464 . . . . . . . 8 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
323adant3 959 . . . . . . 7 ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
4 brinxp 4464 . . . . . . . . 9 ((𝑥𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
543adant2 958 . . . . . . . 8 ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
6 brinxp 4464 . . . . . . . . . 10 ((𝑧𝐴𝑦𝐴) → (𝑧𝑅𝑦𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))
76ancoms 264 . . . . . . . . 9 ((𝑦𝐴𝑧𝐴) → (𝑧𝑅𝑦𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))
873adant1 957 . . . . . . . 8 ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑧𝑅𝑦𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))
95, 8orbi12d 740 . . . . . . 7 ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑧𝑧𝑅𝑦) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦)))
103, 9imbi12d 232 . . . . . 6 ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))))
11103expb 1140 . . . . 5 ((𝑥𝐴 ∧ (𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))))
12112ralbidva 2394 . . . 4 (𝑥𝐴 → (∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ ∀𝑦𝐴𝑧𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))))
1312ralbiia 2386 . . 3 (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦)))
141, 13anbi12i 448 . 2 ((𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))) ↔ ((𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))))
15 df-iso 4088 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
16 df-iso 4088 . 2 ((𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴 ↔ ((𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))))
1714, 15, 163bitr4i 210 1 (𝑅 Or 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wo 662  w3a 920  wcel 1434  wral 2353  cin 2983   class class class wbr 3811   Po wpo 4085   Or wor 4086   × cxp 4399
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-po 4087  df-iso 4088  df-xp 4407
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator