Theorem qsid 6600

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 2741	. . . . . . 7 |- x e. _V
2	1	ecid 6598	. . . . . 6 |- [ x ] `' _E = x
3	2	eqeq2i 2188	. . . . 5 |- ( y = [ x ] `' _E <-> y = x )
4		equcom 1706	. . . . 5 |- ( y = x <-> x = y )
5	3, 4	bitri 184	. . . 4 |- ( y = [ x ] `' _E <-> x = y )
6	5	rexbii 2484	. . 3 |- ( E. x e. A y = [ x ] `' _E <-> E. x e. A x = y )
7		vex 2741	. . . 4 |- y e. _V
8	7	elqs 6586	. . 3 |- ( y e. ( A /. `' _E ) <-> E. x e. A y = [ x ] `' _E )
9		risset 2505	. . 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 2174	1 |- ( A /. `' _E ) = A

Syntax hints:   = wceq 1353   e. wcel 2148   E.wrex 2456   _E cep 4288   `'ccnv 4626   [cec 6533   /.cqs 6534

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-eprel 4290  df-xp 4633  df-cnv 4635  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-ec 6537  df-qs 6541

This theorem is referenced by:  dfcnqs  7840