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Theorem opelco2g 4791
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 4790 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( C  o.  D ) B  <->  E. x ( A D x  /\  x C B ) ) )
2 df-br 4001 . 2  |-  ( A ( C  o.  D
) B  <->  <. A ,  B >.  e.  ( C  o.  D ) )
3 df-br 4001 . . . 4  |-  ( A D x  <->  <. A ,  x >.  e.  D )
4 df-br 4001 . . . 4  |-  ( x C B  <->  <. x ,  B >.  e.  C
)
53, 4anbi12i 460 . . 3  |-  ( ( A D x  /\  x C B )  <->  ( <. A ,  x >.  e.  D  /\  <. x ,  B >.  e.  C ) )
65exbii 1605 . 2  |-  ( E. x ( A D x  /\  x C B )  <->  E. x
( <. A ,  x >.  e.  D  /\  <. x ,  B >.  e.  C
) )
71, 2, 63bitr3g 222 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    /\ wa 104    <-> wb 105   E.wex 1492    e. wcel 2148   <.cop 3594   class class class wbr 4000    o. ccom 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-co 4632
This theorem is referenced by:  dfco2  5124  dmfco  5580
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