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Theorem opelco2g 4980
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 4979 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( C  o.  D ) B  <->  E. x ( A D x  /\  x C B ) ) )
2 df-br 4154 . 2  |-  ( A ( C  o.  D
) B  <->  <. A ,  B >.  e.  ( C  o.  D ) )
3 df-br 4154 . . . 4  |-  ( A D x  <->  <. A ,  x >.  e.  D )
4 df-br 4154 . . . 4  |-  ( x C B  <->  <. x ,  B >.  e.  C
)
53, 4anbi12i 679 . . 3  |-  ( ( A D x  /\  x C B )  <->  ( <. A ,  x >.  e.  D  /\  <. x ,  B >.  e.  C ) )
65exbii 1589 . 2  |-  ( E. x ( A D x  /\  x C B )  <->  E. x
( <. A ,  x >.  e.  D  /\  <. x ,  B >.  e.  C
) )
71, 2, 63bitr3g 279 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 177    /\ wa 359   E.wex 1547    e. wcel 1717   <.cop 3760   class class class wbr 4153    o. ccom 4822
This theorem is referenced by:  dfco2  5309  dmfco  5736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-co 4827
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