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Theorem soinxp 4567
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 4566 . . 3 (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)
2 brinxp 4565 . . . . . . . 8 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
323adant3 982 . . . . . . 7 ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
4 brinxp 4565 . . . . . . . . 9 ((𝑥𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
543adant2 981 . . . . . . . 8 ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
6 brinxp 4565 . . . . . . . . . 10 ((𝑧𝐴𝑦𝐴) → (𝑧𝑅𝑦𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))
76ancoms 266 . . . . . . . . 9 ((𝑦𝐴𝑧𝐴) → (𝑧𝑅𝑦𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))
873adant1 980 . . . . . . . 8 ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑧𝑅𝑦𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))
95, 8orbi12d 765 . . . . . . 7 ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑧𝑧𝑅𝑦) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦)))
103, 9imbi12d 233 . . . . . 6 ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))))
11103expb 1163 . . . . 5 ((𝑥𝐴 ∧ (𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))))
12112ralbidva 2429 . . . 4 (𝑥𝐴 → (∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ ∀𝑦𝐴𝑧𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))))
1312ralbiia 2421 . . 3 (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦)))
141, 13anbi12i 453 . 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 4177 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
16 df-iso 4177 . 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 211 1 (𝑅 Or 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 680  w3a 943  wcel 1461  wral 2388  cin 3034   class class class wbr 3893   Po wpo 4174   Or wor 4175   × cxp 4495
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-po 4176  df-iso 4177  df-xp 4503
This theorem is referenced by: (None)
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