Theorem elqs 6754

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 6753	. 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 1397   e. wcel 2202   E.wrex 2511   _Vcvv 2802   [cec 6699   /.cqs 6700

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-qs 6707

This theorem is referenced by:  qsss  6762  qsid  6768  erovlem  6795  nqnq0  7660