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Theorem qsid 7773
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 3193 . . . . . . 7 𝑥 ∈ V
21ecid 7772 . . . . . 6 [𝑥] E = 𝑥
32eqeq2i 2633 . . . . 5 (𝑦 = [𝑥] E ↔ 𝑦 = 𝑥)
4 equcom 1942 . . . . 5 (𝑦 = 𝑥𝑥 = 𝑦)
53, 4bitri 264 . . . 4 (𝑦 = [𝑥] E ↔ 𝑥 = 𝑦)
65rexbii 3036 . . 3 (∃𝑥𝐴 𝑦 = [𝑥] E ↔ ∃𝑥𝐴 𝑥 = 𝑦)
7 vex 3193 . . . 4 𝑦 ∈ V
87elqs 7759 . . 3 (𝑦 ∈ (𝐴 / E ) ↔ ∃𝑥𝐴 𝑦 = [𝑥] E )
9 risset 3057 . . 3 (𝑦𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝑦)
106, 8, 93bitr4i 292 . 2 (𝑦 ∈ (𝐴 / E ) ↔ 𝑦𝐴)
1110eqriv 2618 1 (𝐴 / E ) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  wrex 2909   E cep 4993  ccnv 5083  [cec 7700   / cqs 7701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-eprel 4995  df-xp 5090  df-cnv 5092  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-ec 7704  df-qs 7708
This theorem is referenced by:  dfcnqs  9923
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