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Theorem qsid 8781
Description: A set is equal to its quotient set modulo the converse membership relation. (Note: the converse membership relation 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 3476 . . . . . . 7 𝑥 ∈ V
21ecid 8780 . . . . . 6 [𝑥] E = 𝑥
32eqeq2i 2743 . . . . 5 (𝑦 = [𝑥] E ↔ 𝑦 = 𝑥)
4 equcom 2019 . . . . 5 (𝑦 = 𝑥𝑥 = 𝑦)
53, 4bitri 274 . . . 4 (𝑦 = [𝑥] E ↔ 𝑥 = 𝑦)
65rexbii 3092 . . 3 (∃𝑥𝐴 𝑦 = [𝑥] E ↔ ∃𝑥𝐴 𝑥 = 𝑦)
7 vex 3476 . . . 4 𝑦 ∈ V
87elqs 8767 . . 3 (𝑦 ∈ (𝐴 / E ) ↔ ∃𝑥𝐴 𝑦 = [𝑥] E )
9 risset 3228 . . 3 (𝑦𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝑦)
106, 8, 93bitr4i 302 . 2 (𝑦 ∈ (𝐴 / E ) ↔ 𝑦𝐴)
1110eqriv 2727 1 (𝐴 / E ) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wcel 2104  wrex 3068   E cep 5580  ccnv 5676  [cec 8705   / cqs 8706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-eprel 5581  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ec 8709  df-qs 8713
This theorem is referenced by:  dfcnqs  11141  cnvepima  37511  n0elim  37825
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