Theorem ecid 6564

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 2729	. . . 4 |- y e. _V
2		ecid.1	. . . 4 |- A e. _V
3	1, 2	elec 6540	. . 3 |- ( y e. [ A ] `' _E <-> A `' _E y )
4	2, 1	brcnv 4787	. . 3 |- ( A `' _E y <-> y _E A )
5	2	epelc 4269	. . 3 |- ( y _E A <-> y e. A )
6	3, 4, 5	3bitri 205	. 2 |- ( y e. [ A ] `' _E <-> y e. A )
7	6	eqriv 2162	1 |- [ A ] `' _E = A

Syntax hints:   = wceq 1343   e. wcel 2136   _Vcvv 2726   class class class wbr 3982   _E cep 4265   `'ccnv 4603   [cec 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-eprel 4267  df-xp 4610  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-ec 6503

This theorem is referenced by:  qsid  6566