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Theorem qsid 6107
 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 2554 . . . . . . 7 x V
21ecid 6105 . . . . . 6 [x] E = x
32eqeq2i 2047 . . . . 5 (y = [x] E ↔ y = x)
4 equcom 1590 . . . . 5 (y = xx = y)
53, 4bitri 173 . . . 4 (y = [x] E ↔ x = y)
65rexbii 2325 . . 3 (x A y = [x] E ↔ x A x = y)
7 vex 2554 . . . 4 y V
87elqs 6093 . . 3 (y (A / E ) ↔ x A y = [x] E )
9 risset 2346 . . 3 (y Ax A x = y)
106, 8, 93bitr4i 201 . 2 (y (A / E ) ↔ y A)
1110eqriv 2034 1 (A / E ) = A
 Colors of variables: wff set class Syntax hints:   = wceq 1242   ∈ wcel 1390  ∃wrex 2301   E cep 4015  ◡ccnv 4287  [cec 6040   / cqs 6041 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-eprel 4017  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-ec 6044  df-qs 6048 This theorem is referenced by:  dfcnqs  6718
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