Theorem elqs 4280

Description: Membership in a quotient set.

Hypothesis

Ref	Expression
elqs.1	|- B e. V

Assertion

Ref	Expression
elqs	|- (B e. (A/.R) <-> E.x(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	. . 3 |- B e. V
2		eqeq1 1478	. . . 4 |- (y = B -> (y = [x]R <-> B = [x]R))
3	2	rexbidv 1661	. . 3 |- (y = B -> (E.x e. A y = [x]R <-> E.x e. A B = [x]R))
4		df-qs 4256	. . 3 |- (A/.R) = {y | E.x e. A y = [x]R}
5	1, 3, 4	elab2 1897	. 2 |- (B e. (A/.R) <-> E.x e. A B = [x]R)
6		df-rex 1647	. 2 |- (E.x e. A B = [x]R <-> E.x(x e. A /\ B = [x]R))
7	5, 6	bitr 173	1 |- (B e. (A/.R) <-> E.x(x e. A /\ B = [x]R))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  E.wex 978  E.wrex 1643  Vcvv 1807  [cec 4249  /.cqs 4250

This theorem is referenced by:  elqsi 4281  ecelqsi 4282  uninqs 10378

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-rex 1647  df-v 1808  df-qs 4256

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