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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 4230 . 2  |-  |^| A  =  { y  |  A. x  e.  A  y  e.  x }
2 df-iin 4274 . 2  |-  |^|_ x  e.  A  x  =  { y  |  A. x  e.  A  y  e.  x }
31, 2eqtr4i 2483 1  |-  |^| A  =  |^|_ x  e.  A  x
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370   {cab 2436   A.wral 2795   |^|cint 4228   |^|_ciin 4272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-ral 2800  df-int 4229  df-iin 4274
This theorem is referenced by:  relint  5063  intpreima  5935  ixpint  7392  firest  14475  efger  16321  rintopn  18640  intcld  18762  iundifdifd  26048  iundifdif  26049
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