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Theorem soinxp 4437
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 4436 . . 3 (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)
2 brinxp 4435 . . . . . . . 8 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
323adant3 935 . . . . . . 7 ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
4 brinxp 4435 . . . . . . . . 9 ((𝑥𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
543adant2 934 . . . . . . . 8 ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
6 brinxp 4435 . . . . . . . . . 10 ((𝑧𝐴𝑦𝐴) → (𝑧𝑅𝑦𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))
76ancoms 259 . . . . . . . . 9 ((𝑦𝐴𝑧𝐴) → (𝑧𝑅𝑦𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))
873adant1 933 . . . . . . . 8 ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑧𝑅𝑦𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))
95, 8orbi12d 717 . . . . . . 7 ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑧𝑧𝑅𝑦) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦)))
103, 9imbi12d 227 . . . . . 6 ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))))
11103expb 1116 . . . . 5 ((𝑥𝐴 ∧ (𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))))
12112ralbidva 2363 . . . 4 (𝑥𝐴 → (∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ ∀𝑦𝐴𝑧𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))))
1312ralbiia 2355 . . 3 (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦)))
141, 13anbi12i 441 . 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 4061 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
16 df-iso 4061 . 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 205 1 (𝑅 Or 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wo 639  w3a 896  wcel 1409  wral 2323  cin 2943   class class class wbr 3791   Po wpo 4058   Or wor 4059   × cxp 4370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-po 4060  df-iso 4061  df-xp 4378
This theorem is referenced by: (None)
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