MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brabga Structured version   Unicode version

Theorem brabga 4677
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 4367 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  R )
2 brabga.2 . . . 4  |-  R  =  { <. x ,  y
>.  |  ph }
32eleq2i 2498 . . 3  |-  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ph } )
41, 3bitri 252 . 2  |-  ( A R B  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ph } )
5 opelopabga.1 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
65opelopabga 4676 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<->  ps ) )
74, 6syl5bb 260 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A R B  <->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   <.cop 3947   class class class wbr 4366   {copab 4424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-br 4367  df-opab 4426
This theorem is referenced by:  braba  4680  brabg  4682  epelg  4708  brcog  4963  fmptco  6015  ofrfval  6497  isfsupp  7840  wemaplem1  8014  oemapval  8140  wemapwe  8154  fpwwe2lem2  9008  fpwwelem  9021  clim  13501  rlim  13502  vdwmc  14871  isstruct2  15073  brssc  15662  isfunc  15712  isfull  15758  isfth  15762  ipole  16347  eqgval  16809  frgpuplem  17365  dvdsr  17817  islindf  19312  ulmval  23277  hpgbr  24744  isuhgra  24967  isushgra  24970  isumgra  24984  isuslgra  25012  isusgra  25013  isausgra  25023  iscusgra  25126  iswlkon  25204  istrlon  25213  ispthon  25248  isspthon  25255  isconngra  25342  isconngra1  25343  erclwwlkeq  25481  erclwwlkneq  25493  iseupa  25635  hlimi  26783  isinftm  28449  metidv  28647  ismntoplly  28781  brae  29016  braew  29017  brfae  29023  climf  37585  isausgr  39009  issubgr  39080  isfusgra  39327  iscllaw  39416  iscomlaw  39417  isasslaw  39419  islininds  39832  lindepsnlininds  39838
  Copyright terms: Public domain W3C validator