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Theorem brab 4419
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 4416 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A R B  <->  ch ) )
71, 2, 6mp2an 654 1  |-  ( A R B  <->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717   _Vcvv 2900   class class class wbr 4154   {copab 4207
This theorem is referenced by:  opbrop  4896  f1oweALT  6014  frxp  6393  fnwelem  6398  dftpos4  6435  dfac3  7936  axdc2lem  8262  brdom7disj  8343  brdom6disj  8344  ordpipq  8753  ltresr  8949  shftfn  11816  2shfti  11823  ex-opab  21589  br8  25138  br6  25139  br4  25140  poseq  25278  dfbigcup2  25464  brcgr  25554  brsegle  25757  heiborlem2  26213
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209
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