Theorem ecelqsg 6554

Description: Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)

Assertion

Ref	Expression
ecelqsg	|- ( ( R e. V /\ B e. A ) -> [ B ] R e. ( A /. R ) )

Proof of Theorem ecelqsg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2165	. . 3 |- [ B ] R = [ B ] R
2		eceq1 6536	. . . . 5 |- ( x = B -> [ x ] R = [ B ] R )
3	2	eqeq2d 2177	. . . 4 |- ( x = B -> ( [ B ] R = [ x ] R <-> [ B ] R = [ B ] R ) )
4	3	rspcev 2830	. . 3 |- ( ( B e. A /\ [ B ] R = [ B ] R ) -> E. x e. A [ B ] R = [ x ] R )
5	1, 4	mpan2 422	. 2 |- ( B e. A -> E. x e. A [ B ] R = [ x ] R )
6		ecexg 6505	. . . 4 |- ( R e. V -> [ B ] R e. _V )
7		elqsg 6551	. . . 4 |- ( [ B ] R e. _V -> ( [ B ] R e. ( A /. R ) <-> E. x e. A [ B ] R = [ x ] R ) )
8	6, 7	syl 14	. . 3 |- ( R e. V -> ( [ B ] R e. ( A /. R ) <-> E. x e. A [ B ] R = [ x ] R ) )
9	8	biimpar 295	. 2 |- ( ( R e. V /\ E. x e. A [ B ] R = [ x ] R ) -> [ B ] R e. ( A /. R ) )
10	5, 9	sylan2 284	1 |- ( ( R e. V /\ B e. A ) -> [ B ] R e. ( A /. R ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   E.wrex 2445   _Vcvv 2726   [cec 6499   /.cqs 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-ec 6503  df-qs 6507

This theorem is referenced by:  ecelqsi  6555  qliftlem  6579  eroprf  6594