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Theorem brco 5173
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 5169 . 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 1596    e. wcel 1767   _Vcvv 3113   class class class wbr 4447    o. ccom 5003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-co 5008
This theorem is referenced by:  opelco  5174  cnvco  5188  resco  5511  imaco  5512  rnco  5513  coass  5526  dffv2  5941  foeqcnvco  6192  f1eqcocnv  6193  imasleval  14799  ustuqtop4  20574  metustexhalfOLD  20893  metustexhalf  20894  rtrclreclem.trans  28820  dftr6  29032  coep  29033  coepr  29034  dfpo2  29037  brtxp  29383  pprodss4v  29387  brpprod  29388  sscoid  29416  elfuns  29418  brimg  29440  brapply  29441  brcup  29442  brcap  29443  brsuccf  29444  funpartlem  29445  brrestrict  29452  dfrdg4  29453  tfrqfree  29454
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