Theorem dfec2 4264

Description: Alternate definition of R-coset of A. Definition 34 of [Suppes] p. 81.

Hypothesis

Ref	Expression
ec2.1	|- A e. V

Assertion

Ref	Expression
dfec2	|- [A]R = {y | ARy}

Distinct variable groups:   y,A   y,R

Proof of Theorem dfec2

Step	Hyp	Ref	Expression
1		dfima3 3406	. 2 |- (R"{A}) = {y | E.x(x e. {A} /\ <.x, y>. e. R)}
2		df-ec 4263	. 2 |- [A]R = (R"{A})
3		ec2.1	. . . . 5 |- A e. V
4		opeq1 2487	. . . . . 6 |- (x = A -> <.x, y>. = <.A, y>.)
5	4	eleq1d 1540	. . . . 5 |- (x = A -> (<.x, y>. e. R <-> <.A, y>. e. R))
6	3, 5	ceqsexv 1835	. . . 4 |- (E.x(x = A /\ <.x, y>. e. R) <-> <.A, y>. e. R)
7		elsn 2421	. . . . . 6 |- (x e. {A} <-> x = A)
8	7	anbi1i 481	. . . . 5 |- ((x e. {A} /\ <.x, y>. e. R) <-> (x = A /\ <.x, y>. e. R))
9	8	exbii 1051	. . . 4 |- (E.x(x e. {A} /\ <.x, y>. e. R) <-> E.x(x = A /\ <.x, y>. e. R))
10		df-br 2620	. . . 4 |- (ARy <-> <.A, y>. e. R)
11	6, 9, 10	3bitr4r 184	. . 3 |- (ARy <-> E.x(x e. {A} /\ <.x, y>. e. R))
12	11	abbii 1575	. 2 |- {y | ARy} = {y | E.x(x e. {A} /\ <.x, y>. e. R)}
13	1, 2, 12	3eqtr4 1505	1 |- [A]R = {y | ARy}

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  Vcvv 1811  {csn 2409  <.cop 2411   class class class wbr 2619  "cima 3173  [cec 4259

This theorem is referenced by:  elec 4279  ecid 4300

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

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