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

Theorem soinxp 4476
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 4475 . . 3 (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)
2 brinxp 4474 . . . . . . . 8 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
323adant3 961 . . . . . . 7 ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
4 brinxp 4474 . . . . . . . . 9 ((𝑥𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
543adant2 960 . . . . . . . 8 ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
6 brinxp 4474 . . . . . . . . . 10 ((𝑧𝐴𝑦𝐴) → (𝑧𝑅𝑦𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))
76ancoms 264 . . . . . . . . 9 ((𝑦𝐴𝑧𝐴) → (𝑧𝑅𝑦𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))
873adant1 959 . . . . . . . 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 1142 . . . . 5 ((𝑥𝐴 ∧ (𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))))
12112ralbidva 2396 . . . 4 (𝑥𝐴 → (∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ ∀𝑦𝐴𝑧𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))))
1312ralbiia 2388 . . 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 4098 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
16 df-iso 4098 . 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 922  wcel 1436  wral 2355  cin 2987   class class class wbr 3820   Po wpo 4095   Or wor 4096   × cxp 4409
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3821  df-opab 3875  df-po 4097  df-iso 4098  df-xp 4417
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator