Theorem qsid 6717

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 2782	. . . . . . 7 ⊢ 𝑥 ∈ V
2	1	ecid 6715	. . . . . 6 ⊢ [𝑥]◡ E = 𝑥
3	2	eqeq2i 2220	. . . . 5 ⊢ (𝑦 = [𝑥]◡ E ↔ 𝑦 = 𝑥)
4		equcom 1732	. . . . 5 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
5	3, 4	bitri 184	. . . 4 ⊢ (𝑦 = [𝑥]◡ E ↔ 𝑥 = 𝑦)
6	5	rexbii 2517	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]◡ E ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦)
7		vex 2782	. . . 4 ⊢ 𝑦 ∈ V
8	7	elqs 6703	. . 3 ⊢ (𝑦 ∈ (𝐴 / ◡ E ) ↔ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]◡ E )
9		risset 2538	. . 3 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦)
10	6, 8, 9	3bitr4i 212	. 2 ⊢ (𝑦 ∈ (𝐴 / ◡ E ) ↔ 𝑦 ∈ 𝐴)
11	10	eqriv 2206	1 ⊢ (𝐴 / ◡ E ) = 𝐴

Syntax hints:   = wceq 1375   ∈ wcel 2180  ∃wrex 2489   E cep 4355  ^◡ccnv 4695  [cec 6648   / cqs 6649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-sbc 3009  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-br 4063  df-opab 4125  df-eprel 4357  df-xp 4702  df-cnv 4704  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-ec 6652  df-qs 6656

This theorem is referenced by:  dfcnqs  7996