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Theorem poinxp 5322
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 5321 . . . . . . . . 9 ((𝑥𝐴𝑥𝐴) → (𝑥𝑅𝑥𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
21anidms 556 . . . . . . . 8 (𝑥𝐴 → (𝑥𝑅𝑥𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
32ad2antrr 705 . . . . . . 7 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → (𝑥𝑅𝑥𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
43notbid 307 . . . . . 6 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
5 brinxp 5321 . . . . . . . . 9 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
65adantr 466 . . . . . . . 8 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
7 brinxp 5321 . . . . . . . . 9 ((𝑦𝐴𝑧𝐴) → (𝑦𝑅𝑧𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧))
87adantll 693 . . . . . . . 8 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → (𝑦𝑅𝑧𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧))
96, 8anbi12d 616 . . . . . . 7 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧)))
10 brinxp 5321 . . . . . . . 8 ((𝑥𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
1110adantlr 694 . . . . . . 7 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → (𝑥𝑅𝑧𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
129, 11imbi12d 333 . . . . . 6 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧)))
134, 12anbi12d 616 . . . . 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 3134 . . . 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 3134 . . 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 3128 . 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 5170 . 2 (𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
18 df-po 5170 . 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 292 1 (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wcel 2145  wral 3061  cin 3722   class class class wbr 4786   Po wpo 5168   × cxp 5247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-po 5170  df-xp 5255
This theorem is referenced by:  soinxp  5323
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