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Theorem soinxp 4708
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 4707 . . 3 (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)
2 brinxp 4706 . . . . . . . 8 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
323adant3 1018 . . . . . . 7 ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
4 brinxp 4706 . . . . . . . . 9 ((𝑥𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
543adant2 1017 . . . . . . . 8 ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
6 brinxp 4706 . . . . . . . . . 10 ((𝑧𝐴𝑦𝐴) → (𝑧𝑅𝑦𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))
76ancoms 268 . . . . . . . . 9 ((𝑦𝐴𝑧𝐴) → (𝑧𝑅𝑦𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))
873adant1 1016 . . . . . . . 8 ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑧𝑅𝑦𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))
95, 8orbi12d 794 . . . . . . 7 ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑧𝑧𝑅𝑦) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦)))
103, 9imbi12d 234 . . . . . 6 ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))))
11103expb 1205 . . . . 5 ((𝑥𝐴 ∧ (𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))))
12112ralbidva 2509 . . . 4 (𝑥𝐴 → (∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ ∀𝑦𝐴𝑧𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))))
1312ralbiia 2501 . . 3 (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦)))
141, 13anbi12i 460 . 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 4309 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
16 df-iso 4309 . 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 212 1 (𝑅 Or 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 709  w3a 979  wcel 2158  wral 2465  cin 3140   class class class wbr 4015   Po wpo 4306   Or wor 4307   × cxp 4636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-po 4308  df-iso 4309  df-xp 4644
This theorem is referenced by: (None)
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