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Theorem soinxp 5144
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 5143 . . 3 (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)
2 brinxp 5142 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
3 biidd 252 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑥 = 𝑦𝑥 = 𝑦))
4 brinxp 5142 . . . . . . 7 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
54ancoms 469 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
62, 3, 53orbi123d 1395 . . . . 5 ((𝑥𝐴𝑦𝐴) → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑥 = 𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
76ralbidva 2979 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑦𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑥 = 𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
87ralbiia 2973 . . 3 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑥 = 𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
91, 8anbi12i 732 . 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 4996 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
11 df-so 4996 . 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 384  w3o 1035  wcel 1987  wral 2907  cin 3554   class class class wbr 4613   Po wpo 4993   Or wor 4994   × cxp 5072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-po 4995  df-so 4996  df-xp 5080
This theorem is referenced by:  weinxp  5147  ltsopi  9654  cnso  14901  opsrtoslem2  19404
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