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Theorem poinxp 4495
Description: Intersection of partial order with cross 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 simpll 496 . . . . . . . 8 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → 𝑥𝐴)
2 brinxp 4494 . . . . . . . 8 ((𝑥𝐴𝑥𝐴) → (𝑥𝑅𝑥𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
31, 1, 2syl2anc 403 . . . . . . 7 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → (𝑥𝑅𝑥𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
43notbid 627 . . . . . 6 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
5 brinxp 4494 . . . . . . . . 9 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
65adantr 270 . . . . . . . 8 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
7 brinxp 4494 . . . . . . . . 9 ((𝑦𝐴𝑧𝐴) → (𝑦𝑅𝑧𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧))
87adantll 460 . . . . . . . 8 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → (𝑦𝑅𝑧𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧))
96, 8anbi12d 457 . . . . . . 7 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧)))
10 brinxp 4494 . . . . . . . 8 ((𝑥𝐴𝑧𝐴) → (𝑥𝑅𝑧𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
1110adantlr 461 . . . . . . 7 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → (𝑥𝑅𝑧𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
129, 11imbi12d 232 . . . . . 6 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝐴) → (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧)))
134, 12anbi12d 457 . . . . 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 2376 . . . 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 2376 . . 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 2392 . 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 4114 . 2 (𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
18 df-po 4114 . 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 210 1 (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wcel 1438  wral 2359  cin 2996   class class class wbr 3837   Po wpo 4112   × cxp 4426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-po 4114  df-xp 4434
This theorem is referenced by:  soinxp  4496
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