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Theorem brabg 4474
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 681 . 2  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ch )
)
4 brabg.5 . 2  |-  R  =  { <. x ,  y
>.  |  ph }
53, 4brabga 4469 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A R B  <->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4212   {copab 4265
This theorem is referenced by:  brab  4477  ideqg  5024  opelcnvg  5052  f1owe  6073  brrpssg  6524  bren  7117  brdomg  7118  brwdom  7535  ltprord  8907  shftfib  11887  efgrelexlema  15381  cmbr  23086  leopg  23625  cvbr  23785  mdbr  23797  dmdbr  23802  soseq  25529  sltval  25602  axcontlem5  25907  isfne  26348  isref  26359  brabg2  26417  isriscg  26600  isfrgra  28380  lcvbr  29819
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 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267
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