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Theorem intiin 4324
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 4228 . 2  |-  |^| A  =  { y  |  A. x  e.  A  y  e.  x }
2 df-iin 4273 . 2  |-  |^|_ x  e.  A  x  =  { y  |  A. x  e.  A  y  e.  x }
31, 2eqtr4i 2434 1  |-  |^| A  =  |^|_ x  e.  A  x
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405   {cab 2387   A.wral 2753   |^|cint 4226   |^|_ciin 4271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-ral 2758  df-int 4227  df-iin 4273
This theorem is referenced by:  relint  4945  intpreima  5995  ixpint  7533  firest  15045  efger  17058  rintopn  19708  intcld  19831  iundifdifd  27845  iundifdif  27846
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