Theorem intiin 3927

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 3833	. 2 |- |^| A = { y | A. x e. A y e. x }
2		df-iin 3876	. 2 |- |^|_ x e. A x = { y | A. x e. A y e. x }
3	1, 2	eqtr4i 2194	1 |- |^| A = |^|_ x e. A x

Syntax hints:   = wceq 1348   {cab 2156   A.wral 2448   |^|cint 3831   |^|_ciin 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-ral 2453  df-int 3832  df-iin 3876

This theorem is referenced by:  relint  4735  ixpintm  6703