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Theorem brcog 5175
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 4456 . . . 4  |-  ( y  =  A  ->  (
y D x  <->  A D x ) )
2 breq2 4457 . . . 4  |-  ( z  =  B  ->  (
x C z  <->  x C B ) )
31, 2bi2anan9 871 . . 3  |-  ( ( y  =  A  /\  z  =  B )  ->  ( ( y D x  /\  x C z )  <->  ( A D x  /\  x C B ) ) )
43exbidv 1690 . 2  |-  ( ( y  =  A  /\  z  =  B )  ->  ( E. x ( y D x  /\  x C z )  <->  E. x
( A D x  /\  x C B ) ) )
5 df-co 5014 . 2  |-  ( C  o.  D )  =  { <. y ,  z
>.  |  E. x
( y D x  /\  x C z ) }
64, 5brabga 4767 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( C  o.  D ) B  <->  E. x ( A D x  /\  x C B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   class class class wbr 4453    o. ccom 5009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-co 5014
This theorem is referenced by:  opelco2g  5176  brcogw  5177  brco  5179  brcodir  5392  brtpos2  6973  ertr  7338  znleval  18460  fcoinvbr  27292  relexpindlem  28894  opelco3  29142  funressnfv  32009
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