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Theorem brcog 4866
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 4042 . . . 4  |-  ( y  =  A  ->  (
y D x  <->  A D x ) )
2 breq2 4043 . . . 4  |-  ( z  =  B  ->  (
x C z  <->  x C B ) )
31, 2bi2anan9 843 . . 3  |-  ( ( y  =  A  /\  z  =  B )  ->  ( ( y D x  /\  x C z )  <->  ( A D x  /\  x C B ) ) )
43exbidv 1616 . 2  |-  ( ( y  =  A  /\  z  =  B )  ->  ( E. x ( y D x  /\  x C z )  <->  E. x
( A D x  /\  x C B ) ) )
5 df-co 4714 . 2  |-  ( C  o.  D )  =  { <. y ,  z
>.  |  E. x
( y D x  /\  x C z ) }
64, 5brabga 4295 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    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   class class class wbr 4039    o. ccom 4709
This theorem is referenced by:  opelco2g  4867  brco  4868  brcodir  5078  brtpos2  6256  ertr  6691  znleval  16524  relexpindlem  24051  funressnfv  28096
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-co 4714
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