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Theorem opelco 4507
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 3765 . 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 4506 . 2  |-  ( A ( C  o.  D
) B  <->  E. x
( A D x  /\  x C B ) )
51, 4bitr3i 175 1  |-  ( <. A ,  B >.  e.  ( C  o.  D
)  <->  E. x ( A D x  /\  x C B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 97    <-> wb 98   E.wex 1381    e. wcel 1393   _Vcvv 2557   <.cop 3378   class class class wbr 3764    o. ccom 4349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-co 4354
This theorem is referenced by:  dmcoss  4601  dmcosseq  4603  cotr  4706  coiun  4830  co02  4834  coi1  4836  coass  4839  fmptco  5330  dftpos4  5878
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