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Theorem poinxp 5756
Description: Intersection of partial order with Cartesian product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
Assertion
Ref Expression
poinxp (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)

Proof of Theorem poinxp
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brinxp 5754 . . . . . . . . 9 ((𝑥𝐴𝑥𝐴) → (𝑥𝑅𝑥𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
21anidms 567 . . . . . . . 8 (𝑥𝐴 → (𝑥𝑅𝑥𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
32ad2antrr 724 . . . . . . 7 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → (𝑥𝑅𝑥𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
43notbid 317 . . . . . 6 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
5 brinxp 5754 . . . . . . . . 9 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
65adantr 481 . . . . . . . 8 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
7 brinxp 5754 . . . . . . . . 9 ((𝑦𝐴𝑧𝐴) → (𝑦𝑅𝑧𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧))
87adantll 712 . . . . . . . 8 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → (𝑦𝑅𝑧𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧))
96, 8anbi12d 631 . . . . . . 7 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧)))
10 brinxp 5754 . . . . . . . 8 ((𝑥𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
1110adantlr 713 . . . . . . 7 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → (𝑥𝑅𝑧𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
129, 11imbi12d 344 . . . . . 6 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧)))
134, 12anbi12d 631 . . . . 5 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ∧ ((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))))
1413ralbidva 3175 . . . 4 ((𝑥𝐴𝑦𝐴) → (∀𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑧𝐴𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ∧ ((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))))
1514ralbidva 3175 . . 3 (𝑥𝐴 → (∀𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑦𝐴𝑧𝐴𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ∧ ((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))))
1615ralbiia 3091 . 2 (∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ∧ ((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧)))
17 df-po 5588 . 2 (𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
18 df-po 5588 . 2 ((𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ∧ ((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧)))
1916, 17, 183bitr4i 302 1 (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wcel 2106  wral 3061  cin 3947   class class class wbr 5148   Po wpo 5586   × 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-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-xp 5682
This theorem is referenced by:  soinxp  5757
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