Theorem ecelqsg 6182

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 2081	. . 3 |- [ B ] R = [ B ] R
2		eceq1 6164	. . . . 5 |- ( x = B -> [ x ] R = [ B ] R )
3	2	eqeq2d 2092	. . . 4 |- ( x = B -> ( [ B ] R = [ x ] R <-> [ B ] R = [ B ] R ) )
4	3	rspcev 2701	. . 3 |- ( ( B e. A /\ [ B ] R = [ B ] R ) -> E. x e. A [ B ] R = [ x ] R )
5	1, 4	mpan2 415	. 2 |- ( B e. A -> E. x e. A [ B ] R = [ x ] R )
6		ecexg 6133	. . . 4 |- ( R e. V -> [ B ] R e. _V )
7		elqsg 6179	. . . 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 291	. 2 |- ( ( R e. V /\ E. x e. A [ B ] R = [ x ] R ) -> [ B ] R e. ( A /. R ) )
10	5, 9	sylan2 280	1 |- ( ( R e. V /\ B e. A ) -> [ B ] R e. ( A /. R ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   E.wrex 2349   _Vcvv 2601   [cec 6127   /.cqs 6128

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-xp 4369  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-ec 6131  df-qs 6135

This theorem is referenced by:  ecelqsi  6183  qliftlem  6207  eroprf  6222