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Theorem brinxp2 5325
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 4844 . 2 (𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵 ↔ (𝐴𝑅𝐵𝐴(𝐶 × 𝐷)𝐵))
2 ancom 465 . 2 ((𝐴𝑅𝐵𝐴(𝐶 × 𝐷)𝐵) ↔ (𝐴(𝐶 × 𝐷)𝐵𝐴𝑅𝐵))
3 brxp 5292 . . . 4 (𝐴(𝐶 × 𝐷)𝐵 ↔ (𝐴𝐶𝐵𝐷))
43anbi1i 733 . . 3 ((𝐴(𝐶 × 𝐷)𝐵𝐴𝑅𝐵) ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝐴𝑅𝐵))
5 df-3an 1074 . . 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 383  w3a 1072  wcel 2127  cin 3702   class class class wbr 4792   × cxp 5252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-br 4793  df-opab 4853  df-xp 5260
This theorem is referenced by:  brinxp  5326  fncnv  6111  erinxp  7976  fpwwe2lem8  9622  fpwwe2lem9  9623  fpwwe2lem12  9626  nqerf  9915  nqerid  9918  isstruct  16043  pwsle  16325  psss  17386  psssdm2  17387  pi1cpbl  23015  pi1grplem  23020
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