Theorem qsid 6768

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 2805	. . . . . . 7 |- x e. _V
2	1	ecid 6766	. . . . . 6 |- [ x ] `' _E = x
3	2	eqeq2i 2242	. . . . 5 |- ( y = [ x ] `' _E <-> y = x )
4		equcom 1754	. . . . 5 |- ( y = x <-> x = y )
5	3, 4	bitri 184	. . . 4 |- ( y = [ x ] `' _E <-> x = y )
6	5	rexbii 2539	. . 3 |- ( E. x e. A y = [ x ] `' _E <-> E. x e. A x = y )
7		vex 2805	. . . 4 |- y e. _V
8	7	elqs 6754	. . 3 |- ( y e. ( A /. `' _E ) <-> E. x e. A y = [ x ] `' _E )
9		risset 2560	. . 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 2228	1 |- ( A /. `' _E ) = A

Syntax hints:   = wceq 1397   e. wcel 2202   E.wrex 2511   _E cep 4384   `'ccnv 4724   [cec 6699   /.cqs 6700

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-eprel 4386  df-xp 4731  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-ec 6703  df-qs 6707

This theorem is referenced by:  dfcnqs  8060