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Theorem qsid 6371
 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

Proof of Theorem qsid
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2623 . . . . . . 7
21ecid 6369 . . . . . 6
32eqeq2i 2099 . . . . 5
4 equcom 1640 . . . . 5
53, 4bitri 183 . . . 4
65rexbii 2386 . . 3
7 vex 2623 . . . 4
87elqs 6357 . . 3
9 risset 2407 . . 3
106, 8, 93bitr4i 211 . 2
1110eqriv 2086 1
 Colors of variables: wff set class Syntax hints:   wceq 1290   wcel 1439  wrex 2361   cep 4123  ccnv 4451  cec 6304  cqs 6305 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045 This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-eprel 4125  df-xp 4458  df-cnv 4460  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-ec 6308  df-qs 6312 This theorem is referenced by:  dfcnqs  7439
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