Theorem qsid 6659

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 2766	. . . . . . 7 |- x e. _V
2	1	ecid 6657	. . . . . 6 |- [ x ] `' _E = x
3	2	eqeq2i 2207	. . . . 5 |- ( y = [ x ] `' _E <-> y = x )
4		equcom 1720	. . . . 5 |- ( y = x <-> x = y )
5	3, 4	bitri 184	. . . 4 |- ( y = [ x ] `' _E <-> x = y )
6	5	rexbii 2504	. . 3 |- ( E. x e. A y = [ x ] `' _E <-> E. x e. A x = y )
7		vex 2766	. . . 4 |- y e. _V
8	7	elqs 6645	. . 3 |- ( y e. ( A /. `' _E ) <-> E. x e. A y = [ x ] `' _E )
9		risset 2525	. . 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 2193	1 |- ( A /. `' _E ) = A

Syntax hints:   = wceq 1364   e. wcel 2167   E.wrex 2476   _E cep 4322   `'ccnv 4662   [cec 6590   /.cqs 6591

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-eprel 4324  df-xp 4669  df-cnv 4671  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-ec 6594  df-qs 6598

This theorem is referenced by:  dfcnqs  7908