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Theorem soinxp 5217
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 5216 . . 3 (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)
2 brinxp 5215 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
3 biidd 252 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑥 = 𝑦𝑥 = 𝑦))
4 brinxp 5215 . . . . . . 7 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
54ancoms 468 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
62, 3, 53orbi123d 1438 . . . . 5 ((𝑥𝐴𝑦𝐴) → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑥 = 𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
76ralbidva 3014 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑦𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑥 = 𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
87ralbiia 3008 . . 3 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑥 = 𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
91, 8anbi12i 733 . 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 5065 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
11 df-so 5065 . 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 1053  wcel 2030  wral 2941  cin 3606   class class class wbr 4685   Po wpo 5062   Or wor 5063   × cxp 5141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-po 5064  df-so 5065  df-xp 5149
This theorem is referenced by:  weinxp  5220  ltsopi  9748  cnso  15020  opsrtoslem2  19533
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