Theorem qsid 8797

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.

Step	Hyp	Ref	Expression
1		vex 3463	. . . . . . 7 ⊢ 𝑥 ∈ V
2	1	ecid 8796	. . . . . 6 ⊢ [𝑥]◡ E = 𝑥
3	2	eqeq2i 2748	. . . . 5 ⊢ (𝑦 = [𝑥]◡ E ↔ 𝑦 = 𝑥)
4		equcom 2017	. . . . 5 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
5	3, 4	bitri 275	. . . 4 ⊢ (𝑦 = [𝑥]◡ E ↔ 𝑥 = 𝑦)
6	5	rexbii 3083	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]◡ E ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦)
7		vex 3463	. . . 4 ⊢ 𝑦 ∈ V
8	7	elqs 8783	. . 3 ⊢ (𝑦 ∈ (𝐴 / ◡ E ) ↔ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]◡ E )
9		risset 3217	. . 3 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦)
10	6, 8, 9	3bitr4i 303	. 2 ⊢ (𝑦 ∈ (𝐴 / ◡ E ) ↔ 𝑦 ∈ 𝐴)
11	10	eqriv 2732	1 ⊢ (𝐴 / ◡ E ) = 𝐴

Syntax hints:   = wceq 1540   ∈ wcel 2108  ∃wrex 3060   E cep 5552  ^◡ccnv 5653  [cec 8717   / cqs 8718

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-eprel 5553  df-xp 5660  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ec 8721  df-qs 8725

This theorem is referenced by:  dfcnqs  11156  cnvepima  38355  n0elim  38668