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Theorem brinxp 5624
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 5623 . 2 (𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵 ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝐴𝑅𝐵))
21baibr 537 1 ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wcel 2105  cin 3934   class class class wbr 5058   × cxp 5547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5059  df-opab 5121  df-xp 5555
This theorem is referenced by:  poinxp  5626  soinxp  5627  frinxp  5628  seinxp  5629  exfo  6864  isores2  7075  ltpiord  10298  ordpinq  10354  pwsleval  16756  tsrss  17823  ordtrest  21740  ordtrest2lem  21741  ordtrestNEW  31064  ordtrest2NEWlem  31065  satefvfmla0  32563
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