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Theorem soinxp 5757
Description: Intersection of total order with Cartesian 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 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poinxp 5756 . . 3 (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)
2 brinxp 5754 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
3 biidd 261 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑥 = 𝑦𝑥 = 𝑦))
4 brinxp 5754 . . . . . . 7 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
54ancoms 459 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
62, 3, 53orbi123d 1435 . . . . 5 ((𝑥𝐴𝑦𝐴) → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑥 = 𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
76ralbidva 3175 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑦𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑥 = 𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
87ralbiia 3091 . . 3 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑥 = 𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
91, 8anbi12i 627 . 2 ((𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ ((𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑥 = 𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
10 df-so 5589 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
11 df-so 5589 . 2 ((𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴 ↔ ((𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑥 = 𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
129, 10, 113bitr4i 302 1 (𝑅 Or 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  w3o 1086  wcel 2106  wral 3061  cin 3947   class class class wbr 5148   Po wpo 5586   Or wor 5587   × cxp 5674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-po 5588  df-so 5589  df-xp 5682
This theorem is referenced by:  weinxp  5760  ltsopi  10882  cnso  16189  opsrtoslem2  21616
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