Theorem qsid 8822

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 3482	. . . . . . 7 ⊢ 𝑥 ∈ V
2	1	ecid 8821	. . . . . 6 ⊢ [𝑥]◡ E = 𝑥
3	2	eqeq2i 2748	. . . . 5 ⊢ (𝑦 = [𝑥]◡ E ↔ 𝑦 = 𝑥)
4		equcom 2015	. . . . 5 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
5	3, 4	bitri 275	. . . 4 ⊢ (𝑦 = [𝑥]◡ E ↔ 𝑥 = 𝑦)
6	5	rexbii 3092	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]◡ E ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦)
7		vex 3482	. . . 4 ⊢ 𝑦 ∈ V
8	7	elqs 8808	. . 3 ⊢ (𝑦 ∈ (𝐴 / ◡ E ) ↔ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]◡ E )
9		risset 3231	. . 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 1537   ∈ wcel 2106  ∃wrex 3068   E cep 5588  ^◡ccnv 5688  [cec 8742   / cqs 8743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-eprel 5589  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746  df-qs 8750

This theorem is referenced by:  dfcnqs  11180  cnvepima  38319  n0elim  38632