Theorem qsid 8530

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 3426	. . . . . . 7 ⊢ 𝑥 ∈ V
2	1	ecid 8529	. . . . . 6 ⊢ [𝑥]◡ E = 𝑥
3	2	eqeq2i 2751	. . . . 5 ⊢ (𝑦 = [𝑥]◡ E ↔ 𝑦 = 𝑥)
4		equcom 2022	. . . . 5 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
5	3, 4	bitri 274	. . . 4 ⊢ (𝑦 = [𝑥]◡ E ↔ 𝑥 = 𝑦)
6	5	rexbii 3177	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]◡ E ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦)
7		vex 3426	. . . 4 ⊢ 𝑦 ∈ V
8	7	elqs 8516	. . 3 ⊢ (𝑦 ∈ (𝐴 / ◡ E ) ↔ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]◡ E )
9		risset 3193	. . 3 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦)
10	6, 8, 9	3bitr4i 302	. 2 ⊢ (𝑦 ∈ (𝐴 / ◡ E ) ↔ 𝑦 ∈ 𝐴)
11	10	eqriv 2735	1 ⊢ (𝐴 / ◡ E ) = 𝐴

Syntax hints:   = wceq 1539   ∈ wcel 2108  ∃wrex 3064   E cep 5485  ^◡ccnv 5579  [cec 8454   / cqs 8455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-eprel 5486  df-xp 5586  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ec 8458  df-qs 8462

This theorem is referenced by:  dfcnqs  10829  cnvepima  36399  n0el3  36690