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Theorem opelco2g 4867
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 4866 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( C  o.  D ) B  <->  E. x ( A D x  /\  x C B ) ) )
2 df-br 4040 . 2  |-  ( A ( C  o.  D
) B  <->  <. A ,  B >.  e.  ( C  o.  D ) )
3 df-br 4040 . . . 4  |-  ( A D x  <->  <. A ,  x >.  e.  D )
4 df-br 4040 . . . 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 1572 . 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 1531    e. wcel 1696   <.cop 3656   class class class wbr 4039    o. ccom 4709
This theorem is referenced by:  dfco2  5188  dmfco  5609
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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