Theorem ecelqsg 5980

Description: Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.)

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 2353	. . 3 |- [B]R = [B]R
2		eceq1 5963	. . . . 5 |- (x = B -> [x]R = [B]R)
3	2	eqeq2d 2364	. . . 4 |- (x = B -> ([B]R = [x]R <-> [B]R = [B]R))
4	3	rspcev 2956	. . 3 |- ((B e. A /\ [B]R = [B]R) -> E.x e. A [B]R = [x]R)
5	1, 4	mpan2 652	. 2 |- (B e. A -> E.x e. A [B]R = [x]R)
6		ecexg 5950	. . . 4 |- (R e. V -> [B]R e. _V)
7		elqsg 5977	. . . 4 |- ([B]R e. _V -> ([B]R e. (A/.R) <-> E.x e. A [B]R = [x]R))
8	6, 7	syl 15	. . 3 |- (R e. V -> ([B]R e. (A/.R) <-> E.x e. A [B]R = [x]R))
9	8	biimpar 471	. 2 |- ((R e. V /\ E.x e. A [B]R = [x]R) -> [B]R e. (A/.R))
10	5, 9	sylan2 460	1 |- ((R e. V /\ B e. A) -> [B]R e. (A/.R))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  E.wrex 2616  _Vcvv 2860  [cec 5946  /.cqs 5947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-br 4641  df-ima 4728  df-ec 5948  df-qs 5952

This theorem is referenced by:  ecelqsi  5981