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Theorem brcog 4674
Description: Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)
Assertion
Ref Expression
brcog  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( C  o.  D ) B  <->  E. x ( A D x  /\  x C B ) ) )
Distinct variable groups:    x, A    x, B    x, C    x, D
Allowed substitution hints:    V( x)    W( x)

Proof of Theorem brcog
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 3900 . . . 4  |-  ( y  =  A  ->  (
y D x  <->  A D x ) )
2 breq2 3901 . . . 4  |-  ( z  =  B  ->  (
x C z  <->  x C B ) )
31, 2bi2anan9 578 . . 3  |-  ( ( y  =  A  /\  z  =  B )  ->  ( ( y D x  /\  x C z )  <->  ( A D x  /\  x C B ) ) )
43exbidv 1779 . 2  |-  ( ( y  =  A  /\  z  =  B )  ->  ( E. x ( y D x  /\  x C z )  <->  E. x
( A D x  /\  x C B ) ) )
5 df-co 4516 . 2  |-  ( C  o.  D )  =  { <. y ,  z
>.  |  E. x
( y D x  /\  x C z ) }
64, 5brabga 4154 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( C  o.  D ) B  <->  E. x ( A D x  /\  x C B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314   E.wex 1451    e. wcel 1463   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:  opelco2g  4675  brcogw  4676  brco  4678  brcodir  4894  foeqcnvco  5657  brtpos2  6114  ertr  6410
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