Theorem qsid 5360

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 5323	. 2 |- (A/.`' _E ) = {y | E.x e. A y = [x]`' _E }
2		visset 2295	. . . . . . . 8 |- x e. _V
3	2	ecid 5359	. . . . . . 7 |- [x]`' _E = x
4	3	eqeq2i 1894	. . . . . 6 |- (y = [x]`' _E <-> y = x)
5		eqcom 1886	. . . . . 6 |- (y = x <-> x = y)
6	4, 5	bitri 190	. . . . 5 |- (y = [x]`' _E <-> x = y)
7	6	rexbii 2128	. . . 4 |- (E.x e. A y = [x]`' _E <-> E.x e. A x = y)
8		risset 2145	. . . 4 |- (y e. A <-> E.x e. A x = y)
9	7, 8	bitr4i 193	. . 3 |- (E.x e. A y = [x]`' _E <-> y e. A)
10	9	abbii 2006	. 2 |- {y | E.x e. A y = [x]`' _E } = {y | y e. A}
11		abid2 2011	. 2 |- {y | y e. A} = A
12	1, 10, 11	3eqtri 1912	1 |- (A/.`' _E ) = A

Syntax hints:   = wceq 1298   e. wcel 1300  {cab 1871  E.wrex 2106   _E cep 3581  `'ccnv 3985  [cec 5316  /.cqs 5317

This theorem is referenced by:  dfcnqs 6414

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-eprel 3583  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-ec 5320  df-qs 5323

Copyright terms: Public domain