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Theorem intiin 4948
 Description: Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)
Assertion
Ref Expression
intiin 𝐴 = 𝑥𝐴 𝑥
Distinct variable group:   𝑥,𝐴

Proof of Theorem intiin
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfint2 4840 . 2 𝐴 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝑥}
2 df-iin 4886 . 2 𝑥𝐴 𝑥 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝑥}
31, 2eqtr4i 2784 1 𝐴 = 𝑥𝐴 𝑥
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  {cab 2735  ∀wral 3070  ∩ cint 4838  ∩ ciin 4884 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-ral 3075  df-int 4839  df-iin 4886 This theorem is referenced by:  trint  5154  relint  5661  intpreima  6829  ixpint  8507  firest  16764  efger  18911  subdrgint  19650  rintopn  21609  intcld  21740  iundifdifd  30423  iundifdif  30424
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