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Theorem brco 4678
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 4674 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A ( C  o.  D ) B  <->  E. x ( A D x  /\  x C B ) ) )
41, 2, 3mp2an 420 1  |-  ( A ( C  o.  D
) B  <->  E. x
( A D x  /\  x C B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104   E.wex 1451    e. wcel 1463   _Vcvv 2658   class class class wbr 3897    o. ccom 4511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-co 4516
This theorem is referenced by:  opelco  4679  cnvco  4692  resco  5011  imaco  5012  rnco  5013  coass  5025  f1eqcocnv  5658
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