Theorem dfin5 3220

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 3219	. 2 |- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
2		df-rab 2531	. 2 |- { x e. A | x e. B } = { x | ( x e. A /\ x e. B ) }
3	1, 2	eqtr4i 2258	1 |- ( A i^i B ) = { x e. A | x e. B }

Syntax hints:   /\ wa 104   = wceq 1398   e. wcel 2205   {cab 2220   {crab 2526   i^i cin 3212

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-gen 1498  ax-4 1559  ax-17 1575  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-cleq 2227  df-rab 2531  df-in 3219

This theorem is referenced by:  nfin  3429  rabbi2dva  3431  ssfidc  7200  2omap  7271  2omapfi  7273  suprzubdc  10600  nninfdcex  10601  nnmindc  12734  nnminle  12735  znnen  13166  bj-inex  16694  pw1map  16786