Theorem dfin5 3227

Description: Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.)

Assertion

Ref	Expression
dfin5	|- ( A i^i B ) = { x e. A | x e. B }

Distinct variable groups:   x, A   x, B

Proof of Theorem dfin5

Step	Hyp	Ref	Expression
1		df-in 3226	. 2 |- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
2		df-rab 2537	. 2 |- { x e. A | x e. B } = { x | ( x e. A /\ x e. B ) }
3	1, 2	eqtr4i 2262	1 |- ( A i^i B ) = { x e. A | x e. B }

This proof depends on syntax axioms:   /\ wa 104   = wceq 1402   e. wcel 2209   {cab 2224   {crab 2532   i^i cin 3219

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231  df-rab 2537  df-in 3226

This theorem is used by:  nfin  3437  rabbi2dva  3439  ssfidc  7249  2omap  7324  2omapfi  7326  suprzubdc  10688  nninfdcex  10689  nnmindc  12836  nnminle  12837  znnen  13347  ppiqub  16293  bj-inex  17138  pw1map  17230