Theorem intiin 4372

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 4277	. 2 |- |^| A = { y | A. x e. A y e. x }
2		df-iin 4321	. 2 |- |^|_ x e. A x = { y | A. x e. A y e. x }
3	1, 2	eqtr4i 2492	1 |- |^| A = |^|_ x e. A x

Syntax hints:   = wceq 1374   {cab 2445   A.wral 2807   |^|cint 4275   |^|_ciin 4319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-ral 2812  df-int 4276  df-iin 4321

This theorem is referenced by:  relint  5117  intpreima  6003  ixpint  7486  firest  14677  efger  16525  rintopn  19178  intcld  19300  iundifdifd  27088  iundifdif  27089