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Theorem soinxp 5340
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 5339 . . 3 (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)
2 brinxp 5338 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
3 biidd 252 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑥 = 𝑦𝑥 = 𝑦))
4 brinxp 5338 . . . . . . 7 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
54ancoms 468 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
62, 3, 53orbi123d 1547 . . . . 5 ((𝑥𝐴𝑦𝐴) → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑥 = 𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
76ralbidva 3123 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑦𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑥 = 𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
87ralbiia 3117 . . 3 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑥 = 𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
91, 8anbi12i 735 . 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 5188 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
11 df-so 5188 . 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 292 1 (𝑅 Or 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3o 1071  wcel 2139  wral 3050  cin 3714   class class class wbr 4804   Po wpo 5185   Or wor 5186   × cxp 5264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-po 5187  df-so 5188  df-xp 5272
This theorem is referenced by:  weinxp  5343  ltsopi  9922  cnso  15195  opsrtoslem2  19707
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