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Theorem brcodir 4896
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 4676 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( `' R  o.  R ) B  <->  E. z ( A R z  /\  z `' R B ) ) )
2 vex 2663 . . . . . 6  |-  z  e. 
_V
3 brcnvg 4690 . . . . . 6  |-  ( ( z  e.  _V  /\  B  e.  W )  ->  ( z `' R B 
<->  B R z ) )
42, 3mpan 420 . . . . 5  |-  ( B  e.  W  ->  (
z `' R B  <-> 
B R z ) )
54anbi2d 459 . . . 4  |-  ( B  e.  W  ->  (
( A R z  /\  z `' R B )  <->  ( A R z  /\  B R z ) ) )
65adantl 275 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A R z  /\  z `' R B )  <->  ( A R z  /\  B R z ) ) )
76exbidv 1781 . 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 187 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    /\ wa 103    <-> wb 104   E.wex 1453    e. wcel 1465   _Vcvv 2660   class class class wbr 3899   `'ccnv 4508    o. ccom 4513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-cnv 4517  df-co 4518
This theorem is referenced by:  codir  4897
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