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Theorem brab 4724
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 4720 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A R B  <->  ch ) )
71, 2, 6mp2an 686 1  |-  ( A R B  <->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1452    e. wcel 1904   _Vcvv 3031   class class class wbr 4395   {copab 4453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455
This theorem is referenced by:  opbrop  4919  f1oweALT  6796  frxp  6925  fnwelem  6930  dftpos4  7010  dfac3  8570  axdc2lem  8896  brdom7disj  8977  brdom6disj  8978  ordpipq  9385  ltresr  9582  shftfn  13213  2shfti  13220  ishpg  24880  brcgr  25009  ex-opab  25961  br8d  28294  br8  30467  br6  30468  br4  30469  poseq  30562  dfbigcup2  30737  brsegle  30946  heiborlem2  32208
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