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Theorem opelco2g 4851
Description: Ordered pair membership in a composition. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 24-Feb-2015.)
Assertion
Ref Expression
opelco2g  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  e.  ( C  o.  D )  <->  E. x
( <. A ,  x >.  e.  D  /\  <. x ,  B >.  e.  C
) ) )
Distinct variable groups:    x, A    x, B    x, C    x, D
Allowed substitution hints:    V( x)    W( x)

Proof of Theorem opelco2g
StepHypRef Expression
1 brcog 4850 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( C  o.  D ) B  <->  E. x ( A D x  /\  x C B ) ) )
2 df-br 4024 . 2  |-  ( A ( C  o.  D
) B  <->  <. A ,  B >.  e.  ( C  o.  D ) )
3 df-br 4024 . . . 4  |-  ( A D x  <->  <. A ,  x >.  e.  D )
4 df-br 4024 . . . 4  |-  ( x C B  <->  <. x ,  B >.  e.  C
)
53, 4anbi12i 678 . . 3  |-  ( ( A D x  /\  x C B )  <->  ( <. A ,  x >.  e.  D  /\  <. x ,  B >.  e.  C ) )
65exbii 1569 . 2  |-  ( E. x ( A D x  /\  x C B )  <->  E. x
( <. A ,  x >.  e.  D  /\  <. x ,  B >.  e.  C
) )
71, 2, 63bitr3g 278 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  e.  ( C  o.  D )  <->  E. x
( <. A ,  x >.  e.  D  /\  <. x ,  B >.  e.  C
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    e. wcel 1684   <.cop 3643   class class class wbr 4023    o. ccom 4693
This theorem is referenced by:  dfco2  5172  dmfco  5593  funpartfv  24483
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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