Theorem elqs 6640

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 6639	. 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 1364   e. wcel 2164   E.wrex 2473   _Vcvv 2760   [cec 6585   /.cqs 6586

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-qs 6593

This theorem is referenced by:  qsss  6648  qsid  6654  erovlem  6681  nqnq0  7501