Theorem qsid 3237

Description: A set is equal to its quotient set mod converse epsilon. (Note: converse epsilon is not an equivalence relation.)

Hypothesis

Ref	Expression
qsid.1	⊢ A ∈ V

Assertion

Ref	Expression
qsid	⊢ (A / ◡E) = A

Proof of Theorem qsid

Step	Hyp	Ref	Expression
1		df-qs 3205	. 2 ⊢ (A / ◡E) = {y∣∃x ∈ A y = [x]◡E}
2		visset 1350	. . . . . . . 8 ⊢ x ∈ V
3	2	ecid 3236	. . . . . . 7 ⊢ [x]◡E = x
4	3	cleq2i 1111	. . . . . 6 ⊢ (y = [x]◡E ↔ y = x)
5		cleqcom 1103	. . . . . 6 ⊢ (y = x ↔ x = y)
6	4, 5	bitr 151	. . . . 5 ⊢ (y = [x]◡E ↔ x = y)
7	6	birex 1224	. . . 4 ⊢ (∃x ∈ A y = [x]◡E ↔ ∃x ∈ A x = y)
8		risset 1235	. . . 4 ⊢ (y ∈ A ↔ ∃x ∈ A x = y)
9	7, 8	bitr4 154	. . 3 ⊢ (∃x ∈ A y = [x]◡E ↔ y ∈ A)
10	9	biabi 1181	. 2 ⊢ {y∣∃x ∈ A y = [x]◡E} = {y∣y ∈ A}
11		abid2 1186	. 2 ⊢ {y∣y ∈ A} = A
12	1, 10, 11	3eqtr 1123	1 ⊢ (A / ◡E) = A

Syntax hints:   = weq 797  {cab 1090   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  Vcvv 1348  Ecep 2056  ^◡ccnv 2409  [cec 3198   / cqs 3199

This theorem is referenced by:  dfcnqs 4056

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-eprel 2122  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-ec 3202  df-qs 3205

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