Theorem qsid 7700

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.

Step	Hyp	Ref	Expression
1		vex 3176	. . . . . . 7 ⊢ 𝑥 ∈ V
2	1	ecid 7699	. . . . . 6 ⊢ [𝑥]◡ E = 𝑥
3	2	eqeq2i 2622	. . . . 5 ⊢ (𝑦 = [𝑥]◡ E ↔ 𝑦 = 𝑥)
4		equcom 1932	. . . . 5 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
5	3, 4	bitri 263	. . . 4 ⊢ (𝑦 = [𝑥]◡ E ↔ 𝑥 = 𝑦)
6	5	rexbii 3023	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]◡ E ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦)
7		vex 3176	. . . 4 ⊢ 𝑦 ∈ V
8	7	elqs 7686	. . 3 ⊢ (𝑦 ∈ (𝐴 / ◡ E ) ↔ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]◡ E )
9		risset 3044	. . 3 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦)
10	6, 8, 9	3bitr4i 291	. 2 ⊢ (𝑦 ∈ (𝐴 / ◡ E ) ↔ 𝑦 ∈ 𝐴)
11	10	eqriv 2607	1 ⊢ (𝐴 / ◡ E ) = 𝐴

Syntax hints:   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   E cep 4947  ^◡ccnv 5037  [cec 7627   / cqs 7628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-eprel 4949  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ec 7631  df-qs 7635

This theorem is referenced by:  dfcnqs  9842