Theorem snec 3232

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 1206	. . . . 5 |- (E.x e. {A}y = [x]R <-> E.x(x e. {A} /\ y = [x]R))
2		elsn 1820	. . . . . . 7 |- (x e. {A} <-> x = A)
3	2	anbi1i 368	. . . . . 6 |- ((x e. {A} /\ y = [x]R) <-> (x = A /\ y = [x]R))
4	3	biex 733	. . . . 5 |- (E.x(x e. {A} /\ y = [x]R) <-> E.x(x = A /\ y = [x]R))
5		snec.1	. . . . . 6 |- A e. V
6		eceq2 3215	. . . . . . 7 |- (x = A -> [x]R = [A]R)
7	6	cleq2d 1112	. . . . . 6 |- (x = A -> (y = [x]R <-> y = [A]R))
8	5, 7	ceqsexv 1371	. . . . 5 |- (E.x(x = A /\ y = [x]R) <-> y = [A]R)
9	1, 4, 8	3bitr 155	. . . 4 |- (E.x e. {A}y = [x]R <-> y = [A]R)
10	9	bicomi 150	. . 3 |- (y = [A]R <-> E.x e. {A}y = [x]R)
11	10	biabi 1181	. 2 |- {y | y = [A]R} = {y | E.x e. {A}y = [x]R}
12		df-sn 1811	. 2 |- {[A]R} = {y | y = [A]R}
13		df-qs 3205	. 2 |- ({A}/.R) = {y | E.x e. {A}y = [x]R}
14	11, 12, 13	3eqtr4 1126	1 |- {[A]R} = ({A}/.R)

Syntax hints:   /\ wa 196  E.wex 678  {cab 1090   = wceq 1091   e. wcel 1092  E.wrex 1202  Vcvv 1348  {csn 1808  [cec 3198  /.cqs 3199

This theorem is referenced by:  ecqs 3233

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-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-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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