Theorem qsid 6705

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 2776	. . . . . . 7 |- x e. _V
2	1	ecid 6703	. . . . . 6 |- [ x ] `' _E = x
3	2	eqeq2i 2217	. . . . 5 |- ( y = [ x ] `' _E <-> y = x )
4		equcom 1730	. . . . 5 |- ( y = x <-> x = y )
5	3, 4	bitri 184	. . . 4 |- ( y = [ x ] `' _E <-> x = y )
6	5	rexbii 2514	. . 3 |- ( E. x e. A y = [ x ] `' _E <-> E. x e. A x = y )
7		vex 2776	. . . 4 |- y e. _V
8	7	elqs 6691	. . 3 |- ( y e. ( A /. `' _E ) <-> E. x e. A y = [ x ] `' _E )
9		risset 2535	. . 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 2203	1 |- ( A /. `' _E ) = A

Syntax hints:   = wceq 1373   e. wcel 2177   E.wrex 2486   _E cep 4347   `'ccnv 4687   [cec 6636   /.cqs 6637

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-br 4055  df-opab 4117  df-eprel 4349  df-xp 4694  df-cnv 4696  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-ec 6640  df-qs 6644

This theorem is referenced by:  dfcnqs  7984