Theorem intiin 4574

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 4477	. 2 |- |^| A = { y | A. x e. A y e. x }
2		df-iin 4523	. 2 |- |^|_ x e. A x = { y | A. x e. A y e. x }
3	1, 2	eqtr4i 2647	1 |- |^| A = |^|_ x e. A x

Syntax hints:   = wceq 1483   {cab 2608   A.wral 2912   |^|cint 4475   |^|_ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-int 4476  df-iin 4523

This theorem is referenced by:  relint  5242  intpreima  6346  ixpint  7935  firest  16093  efger  18131  rintopn  20714  intcld  20844  iundifdifd  29380  iundifdif  29381