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Theorem brabg 4286
Description: The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
opelopabg.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
opelopabg.2  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
brabg.5  |-  R  =  { <. x ,  y
>.  |  ph }
Assertion
Ref Expression
brabg  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( 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)    C( x, y)    D( x, y)    R( x, y)

Proof of Theorem brabg
StepHypRef Expression
1 opelopabg.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
2 opelopabg.2 . . 3  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
31, 2sylan9bb 680 . 2  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ch )
)
4 brabg.5 . 2  |-  R  =  { <. x ,  y
>.  |  ph }
53, 4brabga 4281 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A R B  <->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1625    e. wcel 1686   class class class wbr 4025   {copab 4078
This theorem is referenced by:  brab  4289  ideqg  4837  opelcnvg  4863  f1owe  5852  brrpssg  6281  bren  6873  brdomg  6874  brwdom  7283  ltprord  8656  shftfib  11569  efgrelexlema  15060  cmbr  22165  leopg  22704  cvbr  22864  mdbr  22876  dmdbr  22881  soseq  24256  sltval  24303  axcontlem5  24598  isfne  26279  isref  26290  brabg2  26377  isriscg  26626  isfrgra  28182  lcvbr  29284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-opab 4080
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