Theorem opelco2g 4890

Description: Ordered pair membership in a composition. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 24-Feb-2015.)

Assertion

Ref	Expression
opelco2g	|- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. ( C o. D ) <-> E. x ( <. A , x >. e. D /\ <. x , B >. e. C ) ) )

Distinct variable groups:   x, A   x, B   x, C   x, D

Allowed substitution hints:   V( x)   W( x)

Proof of Theorem opelco2g

Step	Hyp	Ref	Expression
1		brcog 4889	. 2 |- ( ( A e. V /\ B e. W ) -> ( A ( C o. D ) B <-> E. x ( A D x /\ x C B ) ) )
2		df-br 4084	. 2 |- ( A ( C o. D ) B <-> <. A , B >. e. ( C o. D ) )
3		df-br 4084	. . . 4 |- ( A D x <-> <. A , x >. e. D )
4		df-br 4084	. . . 4 |- ( x C B <-> <. x , B >. e. C )
5	3, 4	anbi12i 460	. . 3 |- ( ( A D x /\ x C B ) <-> ( <. A , x >. e. D /\ <. x , B >. e. C ) )
6	5	exbii 1651	. 2 |- ( E. x ( A D x /\ x C B ) <-> E. x ( <. A , x >. e. D /\ <. x , B >. e. C ) )
7	1, 2, 6	3bitr3g 222	1 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. ( C o. D ) <-> E. x ( <. A , x >. e. D /\ <. x , B >. e. C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   E.wex 1538   e. wcel 2200   <.cop 3669   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:  dfco2  5228  dmfco  5702