Theorem ecelqsg 6698

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 2207	. . 3 |- [ B ] R = [ B ] R
2		eceq1 6678	. . . . 5 |- ( x = B -> [ x ] R = [ B ] R )
3	2	eqeq2d 2219	. . . 4 |- ( x = B -> ( [ B ] R = [ x ] R <-> [ B ] R = [ B ] R ) )
4	3	rspcev 2884	. . 3 |- ( ( B e. A /\ [ B ] R = [ B ] R ) -> E. x e. A [ B ] R = [ x ] R )
5	1, 4	mpan2 425	. 2 |- ( B e. A -> E. x e. A [ B ] R = [ x ] R )
6		ecexg 6647	. . . 4 |- ( R e. V -> [ B ] R e. _V )
7		elqsg 6695	. . . 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 297	. 2 |- ( ( R e. V /\ E. x e. A [ B ] R = [ x ] R ) -> [ B ] R e. ( A /. R ) )
10	5, 9	sylan2 286	1 |- ( ( R e. V /\ B e. A ) -> [ B ] R e. ( A /. R ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   E.wrex 2487   _Vcvv 2776   [cec 6641   /.cqs 6642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-xp 4699  df-cnv 4701  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-ec 6645  df-qs 6649

This theorem is referenced by:  ecelqsi  6699  qliftlem  6723  eroprf  6738  quseccl0g  13682