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Theorem qsid 6082
Description: A set is equal to its quotient set mod converse epsilon. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
qsid (A / E ) = A

Proof of Theorem qsid
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2538 . . . . . . 7 x V
21ecid 6080 . . . . . 6 [x] E = x
32eqeq2i 2032 . . . . 5 (y = [x] E ↔ y = x)
4 equcom 1575 . . . . 5 (y = xx = y)
53, 4bitri 173 . . . 4 (y = [x] E ↔ x = y)
65rexbii 2309 . . 3 (x A y = [x] E ↔ x A x = y)
7 vex 2538 . . . 4 y V
87elqs 6068 . . 3 (y (A / E ) ↔ x A y = [x] E )
9 risset 2330 . . 3 (y Ax A x = y)
106, 8, 93bitr4i 201 . 2 (y (A / E ) ↔ y A)
1110eqriv 2019 1 (A / E ) = A
Colors of variables: wff set class
Syntax hints:   = wceq 1228   wcel 1374  wrex 2285   E cep 3998  ccnv 4271  [cec 6015   / cqs 6016
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-eprel 4000  df-xp 4278  df-cnv 4280  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-ec 6019  df-qs 6023
This theorem is referenced by:  dfcnqs  6552
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