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Theorem brcodir 5253
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 5039 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( `' R  o.  R ) B  <->  E. z ( A R z  /\  z `' R B ) ) )
2 vex 2959 . . . . . 6  |-  z  e. 
_V
3 brcnvg 5053 . . . . . 6  |-  ( ( z  e.  _V  /\  B  e.  W )  ->  ( z `' R B 
<->  B R z ) )
42, 3mpan 652 . . . . 5  |-  ( B  e.  W  ->  (
z `' R B  <-> 
B R z ) )
54anbi2d 685 . . . 4  |-  ( B  e.  W  ->  (
( A R z  /\  z `' R B )  <->  ( A R z  /\  B R z ) ) )
65adantl 453 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A R z  /\  z `' R B )  <->  ( A R z  /\  B R z ) ) )
76exbidv 1636 . 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 245 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 177    /\ wa 359   E.wex 1550    e. wcel 1725   _Vcvv 2956   class class class wbr 4212   `'ccnv 4877    o. ccom 4882
This theorem is referenced by:  codir  5254
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 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-cnv 4886  df-co 4887
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