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Theorem soinxp 5767
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 5766 . . 3 (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)
2 brinxp 5764 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
3 biidd 262 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑥 = 𝑦𝑥 = 𝑦))
4 brinxp 5764 . . . . . . 7 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
54ancoms 458 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
62, 3, 53orbi123d 1437 . . . . 5 ((𝑥𝐴𝑦𝐴) → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑥 = 𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
76ralbidva 3176 . . . 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 628 . 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 5593 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
11 df-so 5593 . 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 303 1 (𝑅 Or 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3o 1086  wcel 2108  wral 3061  cin 3950   class class class wbr 5143   Po wpo 5590   Or wor 5591   × cxp 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-po 5592  df-so 5593  df-xp 5691
This theorem is referenced by:  weinxp  5770  ltsopi  10928  cnso  16283  opsrtoslem2  22080
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