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Theorem brcodir 5062
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 4850 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( `' R  o.  R ) B  <->  E. z ( A R z  /\  z `' R B ) ) )
2 vex 2791 . . . . . 6  |-  z  e. 
_V
3 brcnvg 4862 . . . . . 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 1612 . 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 1528    e. wcel 1684   _Vcvv 2788   class class class wbr 4023   `'ccnv 4688    o. ccom 4693
This theorem is referenced by:  codir  5063
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-cnv 4697  df-co 4698
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