Theorem opelco 4649

Description: Ordered pair membership in a composition. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 24-Feb-2015.)

Hypotheses

Ref	Expression
opelco.1	|- A e. _V
opelco.2	|- B e. _V

Assertion

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

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

Proof of Theorem opelco

Step	Hyp	Ref	Expression
1		df-br 3876	. 2 |- ( A ( C o. D ) B <-> <. A , B >. e. ( C o. D ) )
2		opelco.1	. . 3 |- A e. _V
3		opelco.2	. . 3 |- B e. _V
4	2, 3	brco 4648	. 2 |- ( A ( C o. D ) B <-> E. x ( A D x /\ x C B ) )
5	1, 4	bitr3i 185	1 |- ( <. A , B >. e. ( C o. D ) <-> E. x ( A D x /\ x C B ) )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1436   e. wcel 1448   _Vcvv 2641   <.cop 3477   class class class wbr 3875   o. ccom 4481

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-co 4486

This theorem is referenced by:  dmcoss  4744  dmcosseq  4746  cotr  4856  coiun  4984  co02  4988  coi1  4990  coass  4993  fmptco  5518  dftpos4  6090