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Theorem brab 4035
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 4032 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A R B  <->  ch ) )
71, 2, 6mp2an 417 1  |-  ( A R B  <->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285    e. wcel 1434   _Vcvv 2602   class class class wbr 3793   {copab 3846
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 3904  ax-pow 3956  ax-pr 3972
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 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848
This theorem is referenced by:  dftpos4  5912  enq0sym  6684  enq0ref  6685  enq0tr  6686  shftfn  9850
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