Theorem brcog 4889

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 4086	. . . 4 |- ( y = A -> ( y D x <-> A D x ) )
2		breq2 4087	. . . 4 |- ( z = B -> ( x C z <-> x C B ) )
3	1, 2	bi2anan9 608	. . 3 |- ( ( y = A /\ z = B ) -> ( ( y D x /\ x C z ) <-> ( A D x /\ x C B ) ) )
4	3	exbidv 1871	. 2 |- ( ( y = A /\ z = B ) -> ( E. x ( y D x /\ x C z ) <-> E. x ( A D x /\ x C B ) ) )
5		df-co 4728	. 2 |- ( C o. D ) = { <. y , z >. | E. x ( y D x /\ x C z ) }
6	4, 5	brabga 4352	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 104   <-> wb 105   = wceq 1395   E.wex 1538   e. wcel 2200   class class class wbr 4083   o. ccom 4723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-co 4728

This theorem is referenced by:  opelco2g  4890  brcogw  4891  brco  4893  brcodir  5116  foeqcnvco  5914  brtpos2  6397  ertr  6695  znleval  14617