Theorem qsid 4307

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 4272	. 2 |- (A/.`'E) = {y | E.x e. A y = [x]`'E}
2		visset 1816	. . . . . . . 8 |- x e. V
3	2	ecid 4306	. . . . . . 7 |- [x]`'E = x
4	3	eqeq2i 1488	. . . . . 6 |- (y = [x]`'E <-> y = x)
5		eqcom 1480	. . . . . 6 |- (y = x <-> x = y)
6	4, 5	bitr 173	. . . . 5 |- (y = [x]`'E <-> x = y)
7	6	rexbii 1671	. . . 4 |- (E.x e. A y = [x]`'E <-> E.x e. A x = y)
8		risset 1688	. . . 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 1578	. 2 |- {y | E.x e. A y = [x]`'E} = {y | y e. A}
11		abid2 1583	. 2 |- {y | y e. A} = A
12	1, 10, 11	3eqtr 1502	1 |- (A/.`'E) = A

Syntax hints:   = wceq 958   e. wcel 960  {cab 1466  E.wrex 1649  Ecep 2836  `'ccnv 3175  [cec 4265  /.cqs 4266

This theorem is referenced by:  dfcnqs 5274

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-eprel 2838  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-ec 4269  df-qs 4272

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