Theorem qsid 6194

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 2604	. . . . . . 7 |- x e. _V
2	1	ecid 6192	. . . . . 6 |- [ x ] `' _E = x
3	2	eqeq2i 2091	. . . . 5 |- ( y = [ x ] `' _E <-> y = x )
4		equcom 1633	. . . . 5 |- ( y = x <-> x = y )
5	3, 4	bitri 182	. . . 4 |- ( y = [ x ] `' _E <-> x = y )
6	5	rexbii 2373	. . 3 |- ( E. x e. A y = [ x ] `' _E <-> E. x e. A x = y )
7		vex 2604	. . . 4 |- y e. _V
8	7	elqs 6180	. . 3 |- ( y e. ( A /. `' _E ) <-> E. x e. A y = [ x ] `' _E )
9		risset 2394	. . 3 |- ( y e. A <-> E. x e. A x = y )
10	6, 8, 9	3bitr4i 210	. 2 |- ( y e. ( A /. `' _E ) <-> y e. A )
11	10	eqriv 2078	1 |- ( A /. `' _E ) = A

Syntax hints:   = wceq 1284   e. wcel 1433   E.wrex 2349   _E cep 4042   `'ccnv 4362   [cec 6127   /.cqs 6128

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-eprel 4044  df-xp 4369  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-ec 6131  df-qs 6135

This theorem is referenced by:  dfcnqs  7009