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Theorem brinxp2 5151
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 4674 . 2 (𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵 ↔ (𝐴𝑅𝐵𝐴(𝐶 × 𝐷)𝐵))
2 ancom 466 . 2 ((𝐴𝑅𝐵𝐴(𝐶 × 𝐷)𝐵) ↔ (𝐴(𝐶 × 𝐷)𝐵𝐴𝑅𝐵))
3 brxp 5117 . . . 4 (𝐴(𝐶 × 𝐷)𝐵 ↔ (𝐴𝐶𝐵𝐷))
43anbi1i 730 . . 3 ((𝐴(𝐶 × 𝐷)𝐵𝐴𝑅𝐵) ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝐴𝑅𝐵))
5 df-3an 1038 . . 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 1036  wcel 1987  cin 3559   class class class wbr 4623   × cxp 5082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-xp 5090
This theorem is referenced by:  brinxp  5152  fncnv  5930  erinxp  7781  fpwwe2lem8  9419  fpwwe2lem9  9420  fpwwe2lem12  9423  nqerf  9712  nqerid  9715  isstruct  15812  pwsle  16092  psss  17154  psssdm2  17155  pi1cpbl  22784  pi1grplem  22789
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