Theorem opelco 4801

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 4006	. 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 4800	. 2 |- ( A ( C o. D ) B <-> E. x ( A D x /\ x C B ) )
5	1, 4	bitr3i 186	1 |- ( <. A , B >. e. ( C o. D ) <-> E. x ( A D x /\ x C B ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1492   e. wcel 2148   _Vcvv 2739   <.cop 3597   class class class wbr 4005   o. ccom 4632

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-co 4637

This theorem is referenced by:  dmcoss  4898  dmcosseq  4900  cotr  5012  coiun  5140  co02  5144  coi1  5146  coass  5149  fmptco  5684  dftpos4  6266