Theorem brcog 4778

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 3992	. . . 4 |- ( y = A -> ( y D x <-> A D x ) )
2		breq2 3993	. . . 4 |- ( z = B -> ( x C z <-> x C B ) )
3	1, 2	bi2anan9 601	. . 3 |- ( ( y = A /\ z = B ) -> ( ( y D x /\ x C z ) <-> ( A D x /\ x C B ) ) )
4	3	exbidv 1818	. 2 |- ( ( y = A /\ z = B ) -> ( E. x ( y D x /\ x C z ) <-> E. x ( A D x /\ x C B ) ) )
5		df-co 4620	. 2 |- ( C o. D ) = { <. y , z >. | E. x ( y D x /\ x C z ) }
6	4, 5	brabga 4249	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 1348   E.wex 1485   e. wcel 2141   class class class wbr 3989   o. ccom 4615

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-co 4620

This theorem is referenced by:  opelco2g  4779  brcogw  4780  brco  4782  brcodir  4998  foeqcnvco  5769  brtpos2  6230  ertr  6528