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Theorem brinxp 5414
Description: Intersection of binary relation with Cartesian product. (Contributed by NM, 9-Mar-1997.)
Assertion
Ref Expression
brinxp ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵))

Proof of Theorem brinxp
StepHypRef Expression
1 brinxp2 5412 . 2 (𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵 ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝐴𝑅𝐵))
21baibr 534 1 ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wcel 2166  cin 3796   class class class wbr 4872   × cxp 5339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pr 5126
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-br 4873  df-opab 4935  df-xp 5347
This theorem is referenced by:  poinxp  5416  soinxp  5417  frinxp  5418  seinxp  5419  exfo  6625  isores2  6837  ltpiord  10023  ordpinq  10079  pwsleval  16505  tsrss  17575  ordtrest  21376  ordtrest2lem  21377  ordtrestNEW  30511  ordtrest2NEWlem  30512
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