Theorem ecelqsi 3229

Description: Membership of an equivalence class in a quotient set.

Hypothesis

Ref	Expression
ecelqsi.1	|- R e. V

Assertion

Ref	Expression
ecelqsi	|- (B e. A -> [B]R e. (A/.R))

Proof of Theorem ecelqsi

Step	Hyp	Ref	Expression
1		eceq2 3215	. . 3 |- (y = B -> [y]R = [B]R)
2	1	eleq1d 1155	. 2 |- (y = B -> ([y]R e. (A/.R) <-> [B]R e. (A/.R)))
3		a9e 809	. . . 4 |- E.x x = y
4		cleqid 1102	. . . . . 6 |- [y]R = [y]R
5		eleq1 1149	. . . . . . . 8 |- (x = y -> (x e. A <-> y e. A))
6		eceq2 3215	. . . . . . . . 9 |- (x = y -> [x]R = [y]R)
7	6	cleq2d 1112	. . . . . . . 8 |- (x = y -> ([y]R = [x]R <-> [y]R = [y]R))
8	5, 7	anbi12d 476	. . . . . . 7 |- (x = y -> ((x e. A /\ [y]R = [x]R) <-> (y e. A /\ [y]R = [y]R)))
9	8	biimprcd 138	. . . . . 6 |- ((y e. A /\ [y]R = [y]R) -> (x = y -> (x e. A /\ [y]R = [x]R)))
10	4, 9	mpan2 519	. . . . 5 |- (y e. A -> (x = y -> (x e. A /\ [y]R = [x]R)))
11	10	19.22dv 947	. . . 4 |- (y e. A -> (E.x x = y -> E.x(x e. A /\ [y]R = [x]R)))
12	3, 11	mpi 44	. . 3 |- (y e. A -> E.x(x e. A /\ [y]R = [x]R))
13		ecelqsi.1	. . . . 5 |- R e. V
14		ecexg 3204	. . . . 5 |- (R e. V -> [y]R e. V)
15	13, 14	ax-mp 6	. . . 4 |- [y]R e. V
16	15	elqs 3227	. . 3 |- ([y]R e. (A/.R) <-> E.x(x e. A /\ [y]R = [x]R))
17	12, 16	sylibr 175	. 2 |- (y e. A -> [y]R e. (A/.R))
18	2, 17	vtoclga 1387	1 |- (B e. A -> [B]R e. (A/.R))

Syntax hints:   -> wi 2   /\ wa 196  E.wex 678   = weq 797   = wceq 1091   e. wcel 1092  Vcvv 1348  [cec 3198  /.cqs 3199

This theorem is referenced by:  ecopqsi 3230  th3q 3253  1q 3851  addclpq 3852  mulclpq 3854  0r 3983  1r 3984  m1r 3985  addclsr 3986  mulclsr 3987

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-ec 3202  df-qs 3205

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