MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brcog Structured version   Unicode version

Theorem brcog 5006
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 4295 . . . 4  |-  ( y  =  A  ->  (
y D x  <->  A D x ) )
2 breq2 4296 . . . 4  |-  ( z  =  B  ->  (
x C z  <->  x C B ) )
31, 2bi2anan9 868 . . 3  |-  ( ( y  =  A  /\  z  =  B )  ->  ( ( y D x  /\  x C z )  <->  ( A D x  /\  x C B ) ) )
43exbidv 1680 . 2  |-  ( ( y  =  A  /\  z  =  B )  ->  ( E. x ( y D x  /\  x C z )  <->  E. x
( A D x  /\  x C B ) ) )
5 df-co 4849 . 2  |-  ( C  o.  D )  =  { <. y ,  z
>.  |  E. x
( y D x  /\  x C z ) }
64, 5brabga 4603 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 1369   E.wex 1586    e. wcel 1756   class class class wbr 4292    o. ccom 4844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-co 4849
This theorem is referenced by:  opelco2g  5007  brcogw  5008  brco  5010  brcodir  5217  brtpos2  6751  ertr  7116  znleval  17987  relexpindlem  27341  opelco3  27589  funressnfv  30034
  Copyright terms: Public domain W3C validator