Theorem qsid 6686

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 2774	. . . . . . 7 |- x e. _V
2	1	ecid 6684	. . . . . 6 |- [ x ] `' _E = x
3	2	eqeq2i 2215	. . . . 5 |- ( y = [ x ] `' _E <-> y = x )
4		equcom 1728	. . . . 5 |- ( y = x <-> x = y )
5	3, 4	bitri 184	. . . 4 |- ( y = [ x ] `' _E <-> x = y )
6	5	rexbii 2512	. . 3 |- ( E. x e. A y = [ x ] `' _E <-> E. x e. A x = y )
7		vex 2774	. . . 4 |- y e. _V
8	7	elqs 6672	. . 3 |- ( y e. ( A /. `' _E ) <-> E. x e. A y = [ x ] `' _E )
9		risset 2533	. . 3 |- ( y e. A <-> E. x e. A x = y )
10	6, 8, 9	3bitr4i 212	. 2 |- ( y e. ( A /. `' _E ) <-> y e. A )
11	10	eqriv 2201	1 |- ( A /. `' _E ) = A

Syntax hints:   = wceq 1372   e. wcel 2175   E.wrex 2484   _E cep 4333   `'ccnv 4673   [cec 6617   /.cqs 6618

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-eprel 4335  df-xp 4680  df-cnv 4682  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-ec 6621  df-qs 6625

This theorem is referenced by:  dfcnqs  7953