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Theorem qsid 6424
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 2644 . . . . . . 7 𝑥 ∈ V
21ecid 6422 . . . . . 6 [𝑥] E = 𝑥
32eqeq2i 2110 . . . . 5 (𝑦 = [𝑥] E ↔ 𝑦 = 𝑥)
4 equcom 1650 . . . . 5 (𝑦 = 𝑥𝑥 = 𝑦)
53, 4bitri 183 . . . 4 (𝑦 = [𝑥] E ↔ 𝑥 = 𝑦)
65rexbii 2401 . . 3 (∃𝑥𝐴 𝑦 = [𝑥] E ↔ ∃𝑥𝐴 𝑥 = 𝑦)
7 vex 2644 . . . 4 𝑦 ∈ V
87elqs 6410 . . 3 (𝑦 ∈ (𝐴 / E ) ↔ ∃𝑥𝐴 𝑦 = [𝑥] E )
9 risset 2422 . . 3 (𝑦𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝑦)
106, 8, 93bitr4i 211 . 2 (𝑦 ∈ (𝐴 / E ) ↔ 𝑦𝐴)
1110eqriv 2097 1 (𝐴 / E ) = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1299  wcel 1448  wrex 2376   E cep 4147  ccnv 4476  [cec 6357   / cqs 6358
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 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-eprel 4149  df-xp 4483  df-cnv 4485  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-ec 6361  df-qs 6365
This theorem is referenced by:  dfcnqs  7528
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