Theorem intiin 3967

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 3872	. 2 |- |^| A = { y | A. x e. A y e. x }
2		df-iin 3915	. 2 |- |^|_ x e. A x = { y | A. x e. A y e. x }
3	1, 2	eqtr4i 2217	1 |- |^| A = |^|_ x e. A x

Syntax hints:   = wceq 1364   {cab 2179   A.wral 2472   |^|cint 3870   |^|_ciin 3913

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-ral 2477  df-int 3871  df-iin 3915

This theorem is referenced by:  relint  4783  ixpintm  6779