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Theorem brco 5043
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 5039 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A ( C  o.  D ) B  <->  E. x ( A D x  /\  x C B ) ) )
41, 2, 3mp2an 654 1  |-  ( A ( C  o.  D
) B  <->  E. x
( A D x  /\  x C B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1550    e. wcel 1725   _Vcvv 2956   class class class wbr 4212    o. ccom 4882
This theorem is referenced by:  opelco  5044  cnvco  5056  resco  5374  imaco  5375  rnco  5376  coass  5388  dffv2  5796  foeqcnvco  6027  f1eqcocnv  6028  imasleval  13766  ustuqtop4  18274  metustexhalfOLD  18593  metustexhalf  18594  rtrclreclem.trans  25146  dftr6  25373  coep  25374  coepr  25375  dfpo2  25378  brtxp  25725  pprodss4v  25729  brpprod  25730  sscoid  25758  elfuns  25760  brimg  25782  brapply  25783  brcup  25784  brcap  25785  brsuccf  25786  funpartlem  25787  brrestrict  25794  dfrdg4  25795  tfrqfree  25796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-co 4887
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