Theorem snec 4280

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 1642	. . . 4 |- (E.x e. {A}y = [x]R <-> E.x(x e. {A} /\ y = [x]R))
2		elsn 2411	. . . . . 6 |- (x e. {A} <-> x = A)
3	2	anbi1i 480	. . . . 5 |- ((x e. {A} /\ y = [x]R) <-> (x = A /\ y = [x]R))
4	3	exbii 1047	. . . 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 4262	. . . . . 6 |- (x = A -> [x]R = [A]R)
7	6	eqeq2d 1478	. . . . 5 |- (x = A -> (y = [x]R <-> y = [A]R))
8	5, 7	ceqsexv 1826	. . . 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 1567	. 2 |- {y | y = [A]R} = {y | E.x e. {A}y = [x]R}
11		df-sn 2402	. 2 |- {[A]R} = {y | y = [A]R}
12		df-qs 4250	. 2 |- ({A}/.R) = {y | E.x e. {A}y = [x]R}
13	10, 11, 12	3eqtr4 1497	1 |- {[A]R} = ({A}/.R)

Syntax hints:   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  {cab 1456  E.wrex 1638  Vcvv 1802  {csn 2399  [cec 4243  /.cqs 4244

This theorem is referenced by:  ecqs 4281

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-xp 3174  df-cnv 3176  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-ec 4247  df-qs 4250

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