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Theorem qsid 8723
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 3450 . . . . . . 7 𝑥 ∈ V
21ecid 8722 . . . . . 6 [𝑥] E = 𝑥
32eqeq2i 2750 . . . . 5 (𝑦 = [𝑥] E ↔ 𝑦 = 𝑥)
4 equcom 2022 . . . . 5 (𝑦 = 𝑥𝑥 = 𝑦)
53, 4bitri 275 . . . 4 (𝑦 = [𝑥] E ↔ 𝑥 = 𝑦)
65rexbii 3098 . . 3 (∃𝑥𝐴 𝑦 = [𝑥] E ↔ ∃𝑥𝐴 𝑥 = 𝑦)
7 vex 3450 . . . 4 𝑦 ∈ V
87elqs 8709 . . 3 (𝑦 ∈ (𝐴 / E ) ↔ ∃𝑥𝐴 𝑦 = [𝑥] E )
9 risset 3222 . . 3 (𝑦𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝑦)
106, 8, 93bitr4i 303 . 2 (𝑦 ∈ (𝐴 / E ) ↔ 𝑦𝐴)
1110eqriv 2734 1 (𝐴 / E ) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  wrex 3074   E cep 5537  ccnv 5633  [cec 8647   / cqs 8648
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 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
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 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-eprel 5538  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ec 8651  df-qs 8655
This theorem is referenced by:  dfcnqs  11079  cnvepima  36801  n0elim  37115
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