Theorem intiin 4346

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.

Step	Hyp	Ref	Expression
1		dfint2 4250	. 2 |- |^| A = { y | A. x e. A y e. x }
2		df-iin 4295	. 2 |- |^|_ x e. A x = { y | A. x e. A y e. x }
3	1, 2	eqtr4i 2487	1 |- |^| A = |^|_ x e. A x

Syntax hints:   = wceq 1455   {cab 2448   A.wral 2749   |^|cint 4248   |^|_ciin 4293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-ral 2754  df-int 4249  df-iin 4295

This theorem is referenced by:  relint  4976  intpreima  6034  ixpint  7575  firest  15380  efger  17417  rintopn  19988  intcld  20104  iundifdifd  28226  iundifdif  28227