Theorem brcog 4903

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 4096	. . . 4 |- ( y = A -> ( y D x <-> A D x ) )
2		breq2 4097	. . . 4 |- ( z = B -> ( x C z <-> x C B ) )
3	1, 2	bi2anan9 610	. . 3 |- ( ( y = A /\ z = B ) -> ( ( y D x /\ x C z ) <-> ( A D x /\ x C B ) ) )
4	3	exbidv 1873	. 2 |- ( ( y = A /\ z = B ) -> ( E. x ( y D x /\ x C z ) <-> E. x ( A D x /\ x C B ) ) )
5		df-co 4740	. 2 |- ( C o. D ) = { <. y , z >. | E. x ( y D x /\ x C z ) }
6	4, 5	brabga 4364	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 1398   E.wex 1541   e. wcel 2202   class class class wbr 4093   o. ccom 4735

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-co 4740

This theorem is referenced by:  opelco2g  4904  brcogw  4905  brco  4907  brcodir  5131  foeqcnvco  5941  brtpos2  6460  ertr  6760  znleval  14749