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Theorem brco 5022
Description: Binary relation on a composition. (Contributed by NM, 21-Sep-2004.) (Revised by Mario Carneiro, 24-Feb-2015.)
Hypotheses
Ref Expression
opelco.1  |-  A  e. 
_V
opelco.2  |-  B  e. 
_V
Assertion
Ref Expression
brco  |-  ( A ( C  o.  D
) B  <->  E. x
( A D x  /\  x C B ) )
Distinct variable groups:    x, A    x, B    x, C    x, D

Proof of Theorem brco
StepHypRef Expression
1 opelco.1 . 2  |-  A  e. 
_V
2 opelco.2 . 2  |-  B  e. 
_V
3 brcog 5018 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A ( C  o.  D ) B  <->  E. x ( A D x  /\  x C B ) ) )
41, 2, 3mp2an 672 1  |-  ( A ( C  o.  D
) B  <->  E. x
( A D x  /\  x C B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369   E.wex 1586    e. wcel 1756   _Vcvv 2984   class class class wbr 4304    o. ccom 4856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-br 4305  df-opab 4363  df-co 4861
This theorem is referenced by:  opelco  5023  cnvco  5037  resco  5354  imaco  5355  rnco  5356  coass  5368  dffv2  5776  foeqcnvco  6010  f1eqcocnv  6011  imasleval  14491  ustuqtop4  19831  metustexhalfOLD  20150  metustexhalf  20151  rtrclreclem.trans  27360  dftr6  27572  coep  27573  coepr  27574  dfpo2  27577  brtxp  27923  pprodss4v  27927  brpprod  27928  sscoid  27956  elfuns  27958  brimg  27980  brapply  27981  brcup  27982  brcap  27983  brsuccf  27984  funpartlem  27985  brrestrict  27992  dfrdg4  27993  tfrqfree  27994
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