Theorem ecelqsg 6159

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 2040	. . 3 |- [ B ] R = [ B ] R
2		eceq1 6141	. . . . 5 |- ( x = B -> [ x ] R = [ B ] R )
3	2	eqeq2d 2051	. . . 4 |- ( x = B -> ( [ B ] R = [ x ] R <-> [ B ] R = [ B ] R ) )
4	3	rspcev 2656	. . 3 |- ( ( B e. A /\ [ B ] R = [ B ] R ) -> E. x e. A [ B ] R = [ x ] R )
5	1, 4	mpan2 401	. 2 |- ( B e. A -> E. x e. A [ B ] R = [ x ] R )
6		ecexg 6110	. . . 4 |- ( R e. V -> [ B ] R e. _V )
7		elqsg 6156	. . . 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 281	. 2 |- ( ( R e. V /\ E. x e. A [ B ] R = [ x ] R ) -> [ B ] R e. ( A /. R ) )
10	5, 9	sylan2 270	1 |- ( ( R e. V /\ B e. A ) -> [ B ] R e. ( A /. R ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   E.wrex 2307   _Vcvv 2557   [cec 6104   /.cqs 6105

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-ec 6108  df-qs 6112

This theorem is referenced by:  ecelqsi  6160  qliftlem  6184  eroprf  6199