Theorem ecid 6492

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 2689	. . . 4 |- y e. _V
2		ecid.1	. . . 4 |- A e. _V
3	1, 2	elec 6468	. . 3 |- ( y e. [ A ] `' _E <-> A `' _E y )
4	2, 1	brcnv 4722	. . 3 |- ( A `' _E y <-> y _E A )
5	2	epelc 4213	. . 3 |- ( y _E A <-> y e. A )
6	3, 4, 5	3bitri 205	. 2 |- ( y e. [ A ] `' _E <-> y e. A )
7	6	eqriv 2136	1 |- [ A ] `' _E = A

Syntax hints:   = wceq 1331   e. wcel 1480   _Vcvv 2686   class class class wbr 3929   _E cep 4209   `'ccnv 4538   [cec 6427

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-eprel 4211  df-xp 4545  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-ec 6431

This theorem is referenced by:  qsid  6494