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Theorem brabga 4714
Description: The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
opelopabga.1  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
brabga.2  |-  R  =  { <. x ,  y
>.  |  ph }
Assertion
Ref Expression
brabga  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A R B  <->  ps ) )
Distinct variable groups:    x, y, A    x, B, y    ps, x, y
Allowed substitution hints:    ph( x, y)    R( x, y)    V( x, y)    W( x, y)

Proof of Theorem brabga
StepHypRef Expression
1 df-br 4404 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  R )
2 brabga.2 . . . 4  |-  R  =  { <. x ,  y
>.  |  ph }
32eleq2i 2532 . . 3  |-  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ph } )
41, 3bitri 249 . 2  |-  ( A R B  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ph } )
5 opelopabga.1 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
65opelopabga 4713 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<->  ps ) )
74, 6syl5bb 257 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A R B  <->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   <.cop 3994   class class class wbr 4403   {copab 4460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462
This theorem is referenced by:  braba  4717  brabg  4719  epelg  4744  brcog  5117  fmptco  5988  ofrfval  6441  isfsupp  7738  wemaplem1  7874  oemapval  8005  wemapwe  8042  wemapweOLD  8043  fpwwe2lem2  8913  fpwwelem  8926  clim  13093  rlim  13094  vdwmc  14160  isstruct2  14304  brssc  14849  isfunc  14896  isfull  14942  isfth  14946  ipole  15450  eqgval  15852  frgpuplem  16393  dvdsr  16864  islindf  18369  ulmval  21981  legov  23157  isuhgra  23409  isumgra  23421  isuslgra  23443  isusgra  23444  isausgra  23450  iscusgra  23536  iswlkon  23602  istrlon  23612  ispthon  23647  isspthon  23654  isconngra  23730  isconngra1  23731  iseupa  23758  hlimi  24762  fmptcof2  26150  isinftm  26363  metidv  26484  brae  26821  braew  26822  brfae  26828  erclwwlkeq  30649  erclwwlkneq  30665  islininds  31132  lindepsnlininds  31138
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