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Theorem brinxp2 5766
Description: Intersection of binary relation with Cartesian product. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.) Group conjuncts and avoid df-3an 1088. (Revised by Peter Mazsa, 18-Sep-2022.)
Assertion
Ref Expression
brinxp2 (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶𝐴𝐷𝐵) ∧ 𝐶𝑅𝐷))

Proof of Theorem brinxp2
StepHypRef Expression
1 brin 5200 . 2 (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ (𝐶𝑅𝐷𝐶(𝐴 × 𝐵)𝐷))
2 ancom 460 . 2 ((𝐶𝑅𝐷𝐶(𝐴 × 𝐵)𝐷) ↔ (𝐶(𝐴 × 𝐵)𝐷𝐶𝑅𝐷))
3 brxp 5738 . . 3 (𝐶(𝐴 × 𝐵)𝐷 ↔ (𝐶𝐴𝐷𝐵))
43anbi1i 624 . 2 ((𝐶(𝐴 × 𝐵)𝐷𝐶𝑅𝐷) ↔ ((𝐶𝐴𝐷𝐵) ∧ 𝐶𝑅𝐷))
51, 2, 43bitri 297 1 (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶𝐴𝐷𝐵) ∧ 𝐶𝑅𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2106  cin 3962   class class class wbr 5148   × cxp 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695
This theorem is referenced by:  brinxp  5767  opelinxp  5768  fncnv  6641  erinxp  8830  fpwwe2lem7  10675  fpwwe2lem8  10676  fpwwe2lem11  10679  nqerf  10968  nqerid  10971  isstruct  17186  pwsle  17539  psss  18638  psssdm2  18639  pi1cpbl  25091  pi1grplem  25096  br1cnvinxp  38238  brres2  38250  inxpss  38293  inxpss3  38296  idinxpssinxp2  38300  inxp2  38349  inxpxrn  38377
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