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

Theorem intiin 4974
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 4869 . 2 𝐴 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝑥}
2 df-iin 4913 . 2 𝑥𝐴 𝑥 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝑥}
31, 2eqtr4i 2845 1 𝐴 = 𝑥𝐴 𝑥
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  {cab 2797  wral 3136   cint 4867   ciin 4911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-9 2117  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1774  df-sb 2063  df-clab 2798  df-cleq 2812  df-ral 3141  df-int 4868  df-iin 4913
This theorem is referenced by:  trint  5179  relint  5685  intpreima  6831  ixpint  8481  firest  16698  efger  18836  subdrgint  19574  rintopn  21509  intcld  21640  iundifdifd  30305  iundifdif  30306
  Copyright terms: Public domain W3C validator