Theorem elqs 6586

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 6585	. 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 1353   e. wcel 2148   E.wrex 2456   _Vcvv 2738   [cec 6533   /.cqs 6534

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2740  df-qs 6541

This theorem is referenced by:  qsss  6594  qsid  6600  erovlem  6627  nqnq0  7440