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Theorem poinxp 5105
 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 5104 . . . . . . . . 9 ((𝑥𝐴𝑥𝐴) → (𝑥𝑅𝑥𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
21anidms 675 . . . . . . . 8 (𝑥𝐴 → (𝑥𝑅𝑥𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
32ad2antrr 758 . . . . . . 7 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → (𝑥𝑅𝑥𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
43notbid 307 . . . . . 6 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
5 brinxp 5104 . . . . . . . . 9 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
65adantr 480 . . . . . . . 8 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
7 brinxp 5104 . . . . . . . . 9 ((𝑦𝐴𝑧𝐴) → (𝑦𝑅𝑧𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧))
87adantll 746 . . . . . . . 8 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → (𝑦𝑅𝑧𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧))
96, 8anbi12d 743 . . . . . . 7 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧)))
10 brinxp 5104 . . . . . . . 8 ((𝑥𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
1110adantlr 747 . . . . . . 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 743 . . . . 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 2968 . . . 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 2968 . . 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 2962 . 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 4959 . 2 (𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
18 df-po 4959 . 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 291 1 (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  ∀wral 2896   ∩ cin 3539   class class class wbr 4583   Po wpo 4957   × cxp 5036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-po 4959  df-xp 5044 This theorem is referenced by:  soinxp  5106
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