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Theorem intiin 5058
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 4947 . 2 𝐴 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝑥}
2 df-iin 4993 . 2 𝑥𝐴 𝑥 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝑥}
31, 2eqtr4i 2767 1 𝐴 = 𝑥𝐴 𝑥
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  {cab 2713  wral 3060   cint 4945   ciin 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-ral 3061  df-int 4946  df-iin 4993
This theorem is referenced by:  trint  5276  relint  5828  intpreima  7089  ixpint  8966  firest  17478  efger  19737  subdrgint  20805  rintopn  22916  intcld  23049  iundifdifd  32575  iundifdif  32576
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