Theorem opelco2g 4847

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 4846	. 2 |- ( ( A e. V /\ B e. W ) -> ( A ( C o. D ) B <-> E. x ( A D x /\ x C B ) ) )
2		df-br 4046	. 2 |- ( A ( C o. D ) B <-> <. A , B >. e. ( C o. D ) )
3		df-br 4046	. . . 4 |- ( A D x <-> <. A , x >. e. D )
4		df-br 4046	. . . 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 1628	. 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 1515   e. wcel 2176   <.cop 3636   class class class wbr 4045   o. ccom 4680

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-co 4685

This theorem is referenced by:  dfco2  5183  dmfco  5649