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Theorem intiin 3956
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 3864 . 2  |-  |^| A  =  { y  |  A. x  e.  A  y  e.  x }
2 df-iin 3908 . 2  |-  |^|_ x  e.  A  x  =  { y  |  A. x  e.  A  y  e.  x }
31, 2eqtr4i 2306 1  |-  |^| A  =  |^|_ x  e.  A  x
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   |^|cint 3862   |^|_ciin 3906
This theorem is referenced by:  relint  4809  intpreima  5656  ixpint  6843  firest  13337  efger  15027  rintopn  16655  intcld  16777  iundifdifd  23159  iundifdif  23160  inttop3  25516
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-ral 2548  df-int 3863  df-iin 3908
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