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Theorem opelco2g 5032
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 5031 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( C  o.  D ) B  <->  E. x ( A D x  /\  x C B ) ) )
2 df-br 4205 . 2  |-  ( A ( C  o.  D
) B  <->  <. A ,  B >.  e.  ( C  o.  D ) )
3 df-br 4205 . . . 4  |-  ( A D x  <->  <. A ,  x >.  e.  D )
4 df-br 4205 . . . 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 1592 . 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 1550    e. wcel 1725   <.cop 3809   class class class wbr 4204    o. ccom 4874
This theorem is referenced by:  dfco2  5361  dmfco  5789
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-co 4879
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