Theorem qsid 6487

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 2684	. . . . . . 7 |- x e. _V
2	1	ecid 6485	. . . . . 6 |- [ x ] `' _E = x
3	2	eqeq2i 2148	. . . . 5 |- ( y = [ x ] `' _E <-> y = x )
4		equcom 1682	. . . . 5 |- ( y = x <-> x = y )
5	3, 4	bitri 183	. . . 4 |- ( y = [ x ] `' _E <-> x = y )
6	5	rexbii 2440	. . 3 |- ( E. x e. A y = [ x ] `' _E <-> E. x e. A x = y )
7		vex 2684	. . . 4 |- y e. _V
8	7	elqs 6473	. . 3 |- ( y e. ( A /. `' _E ) <-> E. x e. A y = [ x ] `' _E )
9		risset 2461	. . 3 |- ( y e. A <-> E. x e. A x = y )
10	6, 8, 9	3bitr4i 211	. 2 |- ( y e. ( A /. `' _E ) <-> y e. A )
11	10	eqriv 2134	1 |- ( A /. `' _E ) = A

Syntax hints:   = wceq 1331   e. wcel 1480   E.wrex 2415   _E cep 4204   `'ccnv 4533   [cec 6420   /.cqs 6421

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-eprel 4206  df-xp 4540  df-cnv 4542  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-ec 6424  df-qs 6428

This theorem is referenced by:  dfcnqs  7642