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Theorem qsid 8681
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 3448 . . . . . . 7 𝑥 ∈ V
21ecid 8680 . . . . . 6 [𝑥] E = 𝑥
32eqeq2i 2751 . . . . 5 (𝑦 = [𝑥] E ↔ 𝑦 = 𝑥)
4 equcom 2022 . . . . 5 (𝑦 = 𝑥𝑥 = 𝑦)
53, 4bitri 275 . . . 4 (𝑦 = [𝑥] E ↔ 𝑥 = 𝑦)
65rexbii 3096 . . 3 (∃𝑥𝐴 𝑦 = [𝑥] E ↔ ∃𝑥𝐴 𝑥 = 𝑦)
7 vex 3448 . . . 4 𝑦 ∈ V
87elqs 8667 . . 3 (𝑦 ∈ (𝐴 / E ) ↔ ∃𝑥𝐴 𝑦 = [𝑥] E )
9 risset 3220 . . 3 (𝑦𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝑦)
106, 8, 93bitr4i 303 . 2 (𝑦 ∈ (𝐴 / E ) ↔ 𝑦𝐴)
1110eqriv 2735 1 (𝐴 / E ) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  wrex 3072   E cep 5535  ccnv 5631  [cec 8605   / cqs 8606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-eprel 5536  df-xp 5638  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ec 8609  df-qs 8613
This theorem is referenced by:  dfcnqs  11037  cnvepima  36730  n0elim  37044
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