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Theorem brab 4770
Description: The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.)
Hypotheses
Ref Expression
opelopab.1  |-  A  e. 
_V
opelopab.2  |-  B  e. 
_V
opelopab.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
opelopab.4  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
brab.5  |-  R  =  { <. x ,  y
>.  |  ph }
Assertion
Ref Expression
brab  |-  ( A R B  <->  ch )
Distinct variable groups:    x, y, A    x, B, y    ch, x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)    R( x, y)

Proof of Theorem brab
StepHypRef Expression
1 opelopab.1 . 2  |-  A  e. 
_V
2 opelopab.2 . 2  |-  B  e. 
_V
3 opelopab.3 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
4 opelopab.4 . . 3  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
5 brab.5 . . 3  |-  R  =  { <. x ,  y
>.  |  ph }
63, 4, 5brabg 4766 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A R B  <->  ch ) )
71, 2, 6mp2an 672 1  |-  ( A R B  <->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767   _Vcvv 3113   class class class wbr 4447   {copab 4504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506
This theorem is referenced by:  opbrop  5077  f1oweALT  6765  frxp  6890  fnwelem  6895  dftpos4  6971  dfac3  8498  axdc2lem  8824  brdom7disj  8905  brdom6disj  8906  ordpipq  9316  ltresr  9513  shftfn  12865  2shfti  12872  brcgr  23879  ex-opab  24830  br8d  27136  br8  28762  br6  28763  br4  28764  poseq  28910  dfbigcup2  29126  brsegle  29335  heiborlem2  29911
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