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Theorem brcodir 4736
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 4524 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( `' R  o.  R ) B  <->  E. z ( A R z  /\  z `' R B ) ) )
2 vex 2605 . . . . . 6  |-  z  e. 
_V
3 brcnvg 4538 . . . . . 6  |-  ( ( z  e.  _V  /\  B  e.  W )  ->  ( z `' R B 
<->  B R z ) )
42, 3mpan 415 . . . . 5  |-  ( B  e.  W  ->  (
z `' R B  <-> 
B R z ) )
54anbi2d 452 . . . 4  |-  ( B  e.  W  ->  (
( A R z  /\  z `' R B )  <->  ( A R z  /\  B R z ) ) )
65adantl 271 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A R z  /\  z `' R B )  <->  ( A R z  /\  B R z ) ) )
76exbidv 1747 . 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 186 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 102    <-> wb 103   E.wex 1422    e. wcel 1434   _Vcvv 2602   class class class wbr 3787   `'ccnv 4364    o. ccom 4369
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-cnv 4373  df-co 4374
This theorem is referenced by:  codir  4737
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