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Theorem opelco 4853
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 4024 . 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 4852 . 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 1528    e. wcel 1684   _Vcvv 2788   <.cop 3643   class class class wbr 4023    o. ccom 4693
This theorem is referenced by:  dmcoss  4944  dmcosseq  4946  cotr  5055  coiun  5182  co02  5186  coi1  5188  coass  5191  fmptco  5691  dftpos4  6253  fmptcof2  23229  cnvco1  24117  cnvco2  24118  txpss3v  24418  tfrqfree  24489
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-co 4698
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