Theorem intiin 4324

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 4230	. 2 |- |^| A = { y | A. x e. A y e. x }
2		df-iin 4274	. 2 |- |^|_ x e. A x = { y | A. x e. A y e. x }
3	1, 2	eqtr4i 2483	1 |- |^| A = |^|_ x e. A x

Syntax hints:   = wceq 1370   {cab 2436   A.wral 2795   |^|cint 4228   |^|_ciin 4272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-ral 2800  df-int 4229  df-iin 4274

This theorem is referenced by:  relint  5063  intpreima  5935  ixpint  7392  firest  14475  efger  16321  rintopn  18640  intcld  18762  iundifdifd  26048  iundifdif  26049