Theorem ecelqsg 6490

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 2140	. . 3 |- [ B ] R = [ B ] R
2		eceq1 6472	. . . . 5 |- ( x = B -> [ x ] R = [ B ] R )
3	2	eqeq2d 2152	. . . 4 |- ( x = B -> ( [ B ] R = [ x ] R <-> [ B ] R = [ B ] R ) )
4	3	rspcev 2793	. . 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 6441	. . . 4 |- ( R e. V -> [ B ] R e. _V )
7		elqsg 6487	. . . 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 1332   e. wcel 1481   E.wrex 2418   _Vcvv 2689   [cec 6435   /.cqs 6436

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-xp 4553  df-cnv 4555  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-ec 6439  df-qs 6443

This theorem is referenced by:  ecelqsi  6491  qliftlem  6515  eroprf  6530