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Theorem brcodir 5078
Description: Two ways of saying that two elements have an upper bound. (Contributed by Mario Carneiro, 3-Nov-2015.)
Assertion
Ref Expression
brcodir  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( `' R  o.  R ) B  <->  E. z ( A R z  /\  B R z ) ) )
Distinct variable groups:    z, A    z, B    z, R    z, V    z, W

Proof of Theorem brcodir
StepHypRef Expression
1 brcog 4866 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( `' R  o.  R ) B  <->  E. z ( A R z  /\  z `' R B ) ) )
2 vex 2804 . . . . . 6  |-  z  e. 
_V
3 brcnvg 4878 . . . . . 6  |-  ( ( z  e.  _V  /\  B  e.  W )  ->  ( z `' R B 
<->  B R z ) )
42, 3mpan 651 . . . . 5  |-  ( B  e.  W  ->  (
z `' R B  <-> 
B R z ) )
54anbi2d 684 . . . 4  |-  ( B  e.  W  ->  (
( A R z  /\  z `' R B )  <->  ( A R z  /\  B R z ) ) )
65adantl 452 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A R z  /\  z `' R B )  <->  ( A R z  /\  B R z ) ) )
76exbidv 1616 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. z ( A R z  /\  z `' R B )  <->  E. z
( A R z  /\  B R z ) ) )
81, 7bitrd 244 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( `' R  o.  R ) B  <->  E. z ( A R z  /\  B R z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    e. wcel 1696   _Vcvv 2801   class class class wbr 4039   `'ccnv 4704    o. ccom 4709
This theorem is referenced by:  codir  5079
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-cnv 4713  df-co 4714
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