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Theorem brcog 4850
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 4026 . . . 4  |-  ( y  =  A  ->  (
y D x  <->  A D x ) )
2 breq2 4027 . . . 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 1612 . 2  |-  ( ( y  =  A  /\  z  =  B )  ->  ( E. x ( y D x  /\  x C z )  <->  E. x
( A D x  /\  x C B ) ) )
5 df-co 4698 . 2  |-  ( C  o.  D )  =  { <. y ,  z
>.  |  E. x
( y D x  /\  x C z ) }
64, 5brabga 4279 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 1528    = wceq 1623    e. wcel 1684   class class class wbr 4023    o. ccom 4693
This theorem is referenced by:  opelco2g  4851  brco  4852  brcodir  5062  brtpos2  6240  ertr  6675  znleval  16508  relexpindlem  24036  funressnfv  27991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-co 4698
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