Theorem qsid 4301

Description: A set is equal to its quotient set mod converse epsilon. (Note: converse epsilon is not an equivalence relation.)

Assertion

Ref	Expression
qsid	|- (A/.`'E) = A

Proof of Theorem qsid

Step	Hyp	Ref	Expression
1		df-qs 4266	. 2 |- (A/.`'E) = {y | E.x e. A y = [x]`'E}
2		visset 1813	. . . . . . . 8 |- x e. V
3	2	ecid 4300	. . . . . . 7 |- [x]`'E = x
4	3	eqeq2i 1485	. . . . . 6 |- (y = [x]`'E <-> y = x)
5		eqcom 1477	. . . . . 6 |- (y = x <-> x = y)
6	4, 5	bitr 173	. . . . 5 |- (y = [x]`'E <-> x = y)
7	6	rexbii 1668	. . . 4 |- (E.x e. A y = [x]`'E <-> E.x e. A x = y)
8		risset 1685	. . . 4 |- (y e. A <-> E.x e. A x = y)
9	7, 8	bitr4 176	. . 3 |- (E.x e. A y = [x]`'E <-> y e. A)
10	9	abbii 1575	. 2 |- {y | E.x e. A y = [x]`'E} = {y | y e. A}
11		abid2 1580	. 2 |- {y | y e. A} = A
12	1, 10, 11	3eqtr 1499	1 |- (A/.`'E) = A

Syntax hints:   = wceq 956   e. wcel 958  {cab 1463  E.wrex 1646  Ecep 2830  `'ccnv 3169  [cec 4259  /.cqs 4260

This theorem is referenced by:  dfcnqs 5262

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-eprel 2832  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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