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Theorem brco 5162
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 5158 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A ( C  o.  D ) B  <->  E. x ( A D x  /\  x C B ) ) )
41, 2, 3mp2an 670 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 367   E.wex 1617    e. wcel 1823   _Vcvv 3106   class class class wbr 4439    o. ccom 4992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-co 4997
This theorem is referenced by:  opelco  5163  cnvco  5177  resco  5494  imaco  5495  rnco  5496  coass  5509  dffv2  5921  foeqcnvco  6178  f1eqcocnv  6179  rtrclreclem3  12975  imasleval  15030  ustuqtop4  20913  metustexhalfOLD  21232  metustexhalf  21233  dftr6  29420  coep  29421  coepr  29422  dfpo2  29425  brtxp  29758  pprodss4v  29762  brpprod  29763  sscoid  29791  elfuns  29793  brimg  29815  brapply  29816  brcup  29817  brcap  29818  brsuccf  29819  funpartlem  29820  brrestrict  29827  dfrdg4  29828  tfrqfree  29829
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