Theorem ecelqsi 4292

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 4278	. . 3 |- (y = B -> [y]R = [B]R)
2	1	eleq1d 1540	. 2 |- (y = B -> ([y]R e. (A/.R) <-> [B]R e. (A/.R)))
3		a9e 1125	. . . 4 |- E.x x = y
4		eqid 1475	. . . . . 6 |- [y]R = [y]R
5		eleq1 1534	. . . . . . . 8 |- (x = y -> (x e. A <-> y e. A))
6		eceq2 4278	. . . . . . . . 9 |- (x = y -> [x]R = [y]R)
7	6	eqeq2d 1486	. . . . . . . 8 |- (x = y -> ([y]R = [x]R <-> [y]R = [y]R))
8	5, 7	anbi12d 628	. . . . . . 7 |- (x = y -> ((x e. A /\ [y]R = [x]R) <-> (y e. A /\ [y]R = [y]R)))
9	8	biimprcd 156	. . . . . 6 |- ((y e. A /\ [y]R = [y]R) -> (x = y -> (x e. A /\ [y]R = [x]R)))
10	4, 9	mpan2 696	. . . . 5 |- (y e. A -> (x = y -> (x e. A /\ [y]R = [x]R)))
11	10	19.22dv 1290	. . . 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 4265	. . . . 5 |- (R e. V -> [y]R e. V)
15	13, 14	ax-mp 7	. . . 4 |- [y]R e. V
16	15	elqs 4290	. . 3 |- ([y]R e. (A/.R) <-> E.x(x e. A /\ [y]R = [x]R))
17	12, 16	sylibr 200	. 2 |- (y e. A -> [y]R e. (A/.R))
18	2, 17	vtoclga 1852	1 |- (B e. A -> [B]R e. (A/.R))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  Vcvv 1811  [cec 4259  /.cqs 4260

This theorem is referenced by:  ecopqsi 4293  th3q 4317  1q 5057  addclpq 5058  mulclpq 5060  0r 5189  1r 5190  m1r 5191  addclsr 5192  mulclsr 5193

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-ec 4263  df-qs 4266

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