Theorem opelco 4932

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 4115	. 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 4931	. 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 1541   e. wcel 2205   _Vcvv 2815   <.cop 3697   class class class wbr 4114   o. ccom 4758

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-co 4763

This theorem is referenced by:  dmcoss  5032  dmcosseq  5034  cotr  5149  coiun  5277  co02  5281  coi1  5283  coass  5286  fmptco  5848  dftpos4  6507