Theorem brcog 4771

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.

Step	Hyp	Ref	Expression
1		breq1 3985	. . . 4 |- ( y = A -> ( y D x <-> A D x ) )
2		breq2 3986	. . . 4 |- ( z = B -> ( x C z <-> x C B ) )
3	1, 2	bi2anan9 596	. . 3 |- ( ( y = A /\ z = B ) -> ( ( y D x /\ x C z ) <-> ( A D x /\ x C B ) ) )
4	3	exbidv 1813	. 2 |- ( ( y = A /\ z = B ) -> ( E. x ( y D x /\ x C z ) <-> E. x ( A D x /\ x C B ) ) )
5		df-co 4613	. 2 |- ( C o. D ) = { <. y , z >. | E. x ( y D x /\ x C z ) }
6	4, 5	brabga 4242	1 |- ( ( A e. V /\ B e. W ) -> ( A ( C o. D ) B <-> E. x ( A D x /\ x C B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   class class class wbr 3982   o. ccom 4608

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-co 4613

This theorem is referenced by:  opelco2g  4772  brcogw  4773  brco  4775  brcodir  4991  foeqcnvco  5758  brtpos2  6219  ertr  6516