Theorem intiin 4046

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 3951	. 2 |- |^| A = { y | A. x e. A y e. x }
2		df-iin 3994	. 2 |- |^|_ x e. A x = { y | A. x e. A y e. x }
3	1, 2	eqtr4i 2256	1 |- |^| A = |^|_ x e. A x

Syntax hints:   = wceq 1398   {cab 2218   A.wral 2520   |^|cint 3949   |^|_ciin 3992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-ral 2525  df-int 3950  df-iin 3994

This theorem is referenced by:  relint  4876  ixpintm  6960