Theorem elqs 6798

Description: Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)

Hypothesis

Ref	Expression
elqs.1	|- B e. _V

Assertion

Ref	Expression
elqs	|- ( B e. ( A /. R ) <-> E. x e. A B = [ x ] R )

Distinct variable groups:   x, A   x, B   x, R

Proof of Theorem elqs

Step	Hyp	Ref	Expression
1		elqs.1	. 2 |- B e. _V
2		elqsg 6797	. 2 |- ( B e. _V -> ( B e. ( A /. R ) <-> E. x e. A B = [ x ] R ) )
3	1, 2	ax-mp 5	1 |- ( B e. ( A /. R ) <-> E. x e. A B = [ x ] R )

Syntax hints:   <-> wb 105   = wceq 1398   e. wcel 2202   E.wrex 2512   _Vcvv 2803   [cec 6743   /.cqs 6744

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-qs 6751

This theorem is referenced by:  qsss  6806  qsid  6812  erovlem  6839  nqnq0  7721