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Theorem brinxp2 5178
Description: Intersection of binary relation with Cartesian product. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
brinxp2 (𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵 ↔ (𝐴𝐶𝐵𝐷𝐴𝑅𝐵))

Proof of Theorem brinxp2
StepHypRef Expression
1 brin 4702 . 2 (𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵 ↔ (𝐴𝑅𝐵𝐴(𝐶 × 𝐷)𝐵))
2 ancom 466 . 2 ((𝐴𝑅𝐵𝐴(𝐶 × 𝐷)𝐵) ↔ (𝐴(𝐶 × 𝐷)𝐵𝐴𝑅𝐵))
3 brxp 5145 . . . 4 (𝐴(𝐶 × 𝐷)𝐵 ↔ (𝐴𝐶𝐵𝐷))
43anbi1i 731 . . 3 ((𝐴(𝐶 × 𝐷)𝐵𝐴𝑅𝐵) ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝐴𝑅𝐵))
5 df-3an 1039 . . 3 ((𝐴𝐶𝐵𝐷𝐴𝑅𝐵) ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝐴𝑅𝐵))
64, 5bitr4i 267 . 2 ((𝐴(𝐶 × 𝐷)𝐵𝐴𝑅𝐵) ↔ (𝐴𝐶𝐵𝐷𝐴𝑅𝐵))
71, 2, 63bitri 286 1 (𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵 ↔ (𝐴𝐶𝐵𝐷𝐴𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3a 1037  wcel 1989  cin 3571   class class class wbr 4651   × cxp 5110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-xp 5118
This theorem is referenced by:  brinxp  5179  fncnv  5960  erinxp  7818  fpwwe2lem8  9456  fpwwe2lem9  9457  fpwwe2lem12  9460  nqerf  9749  nqerid  9752  isstruct  15864  pwsle  16146  psss  17208  psssdm2  17209  pi1cpbl  22838  pi1grplem  22843
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