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Theorem opelco 4869
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
StepHypRef Expression
1 df-br 4040 . 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
42, 3brco 4868 . 2  |-  ( A ( C  o.  D
) B  <->  E. x
( A D x  /\  x C B ) )
51, 4bitr3i 242 1  |-  ( <. A ,  B >.  e.  ( C  o.  D
)  <->  E. x ( A D x  /\  x C B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1531    e. wcel 1696   _Vcvv 2801   <.cop 3656   class class class wbr 4039    o. ccom 4709
This theorem is referenced by:  dmcoss  4960  dmcosseq  4962  cotr  5071  coiun  5198  co02  5202  coi1  5204  coass  5207  fmptco  5707  dftpos4  6269  fmptcof2  23244  cnvco1  24188  cnvco2  24189  txpss3v  24489  tfrqfree  24561
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-co 4714
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