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Theorem brab 4469
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 4466 . 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 1652    e. wcel 1725   _Vcvv 2948   class class class wbr 4204   {copab 4257
This theorem is referenced by:  opbrop  4947  f1oweALT  6066  frxp  6448  fnwelem  6453  dftpos4  6490  dfac3  7994  axdc2lem  8320  brdom7disj  8401  brdom6disj  8402  ordpipq  8811  ltresr  9007  shftfn  11880  2shfti  11887  ex-opab  21732  br8  25371  br6  25372  br4  25373  poseq  25520  dfbigcup2  25736  brcgr  25831  brsegle  26034  heiborlem2  26512
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259
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