Theorem ecid 6192

Description: A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Hypothesis

Ref	Expression
ecid.1	|- A e. _V

Assertion

Ref	Expression
ecid	|- [ A ] `' _E = A

Proof of Theorem ecid

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2604	. . . 4 |- y e. _V
2		ecid.1	. . . 4 |- A e. _V
3	1, 2	elec 6168	. . 3 |- ( y e. [ A ] `' _E <-> A `' _E y )
4	2, 1	brcnv 4536	. . 3 |- ( A `' _E y <-> y _E A )
5	2	epelc 4046	. . 3 |- ( y _E A <-> y e. A )
6	3, 4, 5	3bitri 204	. 2 |- ( y e. [ A ] `' _E <-> y e. A )
7	6	eqriv 2078	1 |- [ A ] `' _E = A

Syntax hints:   = wceq 1284   e. wcel 1433   _Vcvv 2601   class class class wbr 3785   _E cep 4042   `'ccnv 4362   [cec 6127

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

This theorem is referenced by:  qsid  6194