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Theorem brabg 4756
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 699 . 2  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ch )
)
4 brabg.5 . 2  |-  R  =  { <. x ,  y
>.  |  ph }
53, 4brabga 4751 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A R B  <->  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   class class class wbr 4437   {copab 4494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496
This theorem is referenced by:  brab  4760  ideqg  5144  opelcnvg  5172  f1owe  6234  brrpssg  6567  bren  7527  brdomg  7528  brwdom  7996  ltprord  9411  shftfib  12886  efgrelexlema  16745  isref  19987  istrkgld  23833  islnopp  24089  axcontlem5  24247  isfrgra  24966  cmbr  26478  leopg  27017  cvbr  27177  mdbr  27189  dmdbr  27194  soseq  29309  sltval  29382  isfne  30132  brabg2  30181  isriscg  30362  lcvbr  34486
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