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

Theorem inteq 4045
Description: Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
Assertion
Ref Expression
inteq  |-  ( A  =  B  ->  |^| A  =  |^| B )

Proof of Theorem inteq
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 raleq 2896 . . 3  |-  ( A  =  B  ->  ( A. y  e.  A  x  e.  y  <->  A. y  e.  B  x  e.  y ) )
21abbidv 2549 . 2  |-  ( A  =  B  ->  { x  |  A. y  e.  A  x  e.  y }  =  { x  |  A. y  e.  B  x  e.  y } )
3 dfint2 4044 . 2  |-  |^| A  =  { x  |  A. y  e.  A  x  e.  y }
4 dfint2 4044 . 2  |-  |^| B  =  { x  |  A. y  e.  B  x  e.  y }
52, 3, 43eqtr4g 2492 1  |-  ( A  =  B  ->  |^| A  =  |^| B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   {cab 2421   A.wral 2697   |^|cint 4042
This theorem is referenced by:  inteqi  4046  inteqd  4047  unissint  4066  uniintsn  4079  rint0  4082  intex  4348  intnex  4349  elreldm  5086  elxp5  5350  1stval2  6356  oev2  6759  fundmen  7172  xpsnen  7184  fiint  7375  elfir  7412  inelfi  7415  fiin  7419  cardmin2  7877  isfin2-2  8191  incexclem  12608  xpnnenOLD  12801  mreintcl  13812  ismred2  13820  fiinopn  16966  cmpfii  17464  ptbasfi  17605  fbssint  17862  shintcl  22824  chintcl  22826  rankeq1o  26104  neificl  26450  heibor1lem  26509  elrfi  26739  elrfirn  26740
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-int 4043
  Copyright terms: Public domain W3C validator