Theorem snec 4296

Description: The singleton of an equivalence class.

Hypothesis

Ref	Expression
snec.1	|- A e. V

Assertion

Ref	Expression
snec	|- {[A]R} = ({A}/.R)

Proof of Theorem snec

Step	Hyp	Ref	Expression
1		df-rex 1650	. . . 4 |- (E.x e. {A}y = [x]R <-> E.x(x e. {A} /\ y = [x]R))
2		elsn 2421	. . . . . 6 |- (x e. {A} <-> x = A)
3	2	anbi1i 481	. . . . 5 |- ((x e. {A} /\ y = [x]R) <-> (x = A /\ y = [x]R))
4	3	exbii 1051	. . . 4 |- (E.x(x e. {A} /\ y = [x]R) <-> E.x(x = A /\ y = [x]R))
5		snec.1	. . . . 5 |- A e. V
6		eceq2 4278	. . . . . 6 |- (x = A -> [x]R = [A]R)
7	6	eqeq2d 1486	. . . . 5 |- (x = A -> (y = [x]R <-> y = [A]R))
8	5, 7	ceqsexv 1835	. . . 4 |- (E.x(x = A /\ y = [x]R) <-> y = [A]R)
9	1, 4, 8	3bitrr 178	. . 3 |- (y = [A]R <-> E.x e. {A}y = [x]R)
10	9	abbii 1575	. 2 |- {y | y = [A]R} = {y | E.x e. {A}y = [x]R}
11		df-sn 2412	. 2 |- {[A]R} = {y | y = [A]R}
12		df-qs 4266	. 2 |- ({A}/.R) = {y | E.x e. {A}y = [x]R}
13	10, 11, 12	3eqtr4 1505	1 |- {[A]R} = ({A}/.R)

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  E.wrex 1646  Vcvv 1811  {csn 2409  [cec 4259  /.cqs 4260

This theorem is referenced by:  ecqs 4297

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-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

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-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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