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

Theorem intiin 3972
Description: Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)
Assertion
Ref Expression
intiin  |-  |^| A  =  |^|_ x  e.  A  x
Distinct variable group:    x, A

Proof of Theorem intiin
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfint2 3880 . 2  |-  |^| A  =  { y  |  A. x  e.  A  y  e.  x }
2 df-iin 3924 . 2  |-  |^|_ x  e.  A  x  =  { y  |  A. x  e.  A  y  e.  x }
31, 2eqtr4i 2319 1  |-  |^| A  =  |^|_ x  e.  A  x
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   |^|cint 3878   |^|_ciin 3922
This theorem is referenced by:  relint  4825  intpreima  5672  ixpint  6859  firest  13353  efger  15043  rintopn  16671  intcld  16793  iundifdifd  23175  iundifdif  23176  inttop3  25619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-ral 2561  df-int 3879  df-iin 3924
  Copyright terms: Public domain W3C validator