Theorem qsid 6371

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 2623	. . . . . . 7 |- x e. _V
2	1	ecid 6369	. . . . . 6 |- [ x ] `' _E = x
3	2	eqeq2i 2099	. . . . 5 |- ( y = [ x ] `' _E <-> y = x )
4		equcom 1640	. . . . 5 |- ( y = x <-> x = y )
5	3, 4	bitri 183	. . . 4 |- ( y = [ x ] `' _E <-> x = y )
6	5	rexbii 2386	. . 3 |- ( E. x e. A y = [ x ] `' _E <-> E. x e. A x = y )
7		vex 2623	. . . 4 |- y e. _V
8	7	elqs 6357	. . 3 |- ( y e. ( A /. `' _E ) <-> E. x e. A y = [ x ] `' _E )
9		risset 2407	. . 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 2086	1 |- ( A /. `' _E ) = A

Syntax hints:   = wceq 1290   e. wcel 1439   E.wrex 2361   _E cep 4123   `'ccnv 4451   [cec 6304   /.cqs 6305

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-eprel 4125  df-xp 4458  df-cnv 4460  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-ec 6308  df-qs 6312

This theorem is referenced by:  dfcnqs  7439