Theorem intiin 3920

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 3826	. 2 |- |^| A = { y | A. x e. A y e. x }
2		df-iin 3869	. 2 |- |^|_ x e. A x = { y | A. x e. A y e. x }
3	1, 2	eqtr4i 2189	1 |- |^| A = |^|_ x e. A x

Syntax hints:   = wceq 1343   {cab 2151   A.wral 2444   |^|cint 3824   |^|_ciin 3867

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-ral 2449  df-int 3825  df-iin 3869

This theorem is referenced by:  relint  4728  ixpintm  6691