Theorem qsid 6502

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 2692	. . . . . . 7 |- x e. _V
2	1	ecid 6500	. . . . . 6 |- [ x ] `' _E = x
3	2	eqeq2i 2151	. . . . 5 |- ( y = [ x ] `' _E <-> y = x )
4		equcom 1683	. . . . 5 |- ( y = x <-> x = y )
5	3, 4	bitri 183	. . . 4 |- ( y = [ x ] `' _E <-> x = y )
6	5	rexbii 2445	. . 3 |- ( E. x e. A y = [ x ] `' _E <-> E. x e. A x = y )
7		vex 2692	. . . 4 |- y e. _V
8	7	elqs 6488	. . 3 |- ( y e. ( A /. `' _E ) <-> E. x e. A y = [ x ] `' _E )
9		risset 2466	. . 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 2137	1 |- ( A /. `' _E ) = A

Syntax hints:   = wceq 1332   e. wcel 1481   E.wrex 2418   _E cep 4217   `'ccnv 4546   [cec 6435   /.cqs 6436

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-eprel 4219  df-xp 4553  df-cnv 4555  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-ec 6439  df-qs 6443

This theorem is referenced by:  dfcnqs  7673