Theorem opelco2g 4946

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 4945	. 2 |- ( ( A e. V /\ B e. W ) -> ( A ( C o. D ) B <-> E. x ( A D x /\ x C B ) ) )
2		df-br 4129	. 2 |- ( A ( C o. D ) B <-> <. A , B >. e. ( C o. D ) )
3		df-br 4129	. . . 4 |- ( A D x <-> <. A , x >. e. D )
4		df-br 4129	. . . 4 |- ( x C B <-> <. x , B >. e. C )
5	3, 4	anbi12i 464	. . 3 |- ( ( A D x /\ x C B ) <-> ( <. A , x >. e. D /\ <. x , B >. e. C ) )
6	5	exbii 1658	. 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 1545   e. wcel 2209   <.cop 3711   class class class wbr 4128   o. ccom 4776

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-co 4781

This theorem is referenced by:  dfco2  5285  dmfco  5770